Part I, where I go over the official Church teaching.
Part III, where I go over the teaching of the Church Fathers.
Part II: The Papacy: A Look at the Biblical Case and the Apostolic Tradition of St. Peter, Vicar of Christ.
Now, let us go through Sacred Scripture for the elements of this doctrine that find their root there.
“And Jesus answered him, “Blessed are you, Simon son of Jonah! For flesh and blood has not revealed this to you, but my Father in heaven. And I tell you, you are Peter, and on this rock I will build my church, and the gates of Hades will not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.’ (Matthew 16:17-19)” This is one of the strongest Petrine text that biblically shows the doctrine of the papacy, let’s unpack it.
“…‘Blessed are you, Simon son of Jonah! For flesh and blood has not revealed this to you, but my Father in heaven. (Matthew 16:17)” Here, St Peter is being singled out by Christ, while regularly he speaks for the rest of the apostles, this is not that, Christ assures us. This is a revelation solely from the Father , blessed solely upon St. Peter.
“And I tell you, you are Peter, and on this rock I will build my church, and the gates of Hades will not prevail against it. (Matthew 16:18)” Peter is Greek for rock and so the sentence should be read as “you are Rock, upon this rock…”making it clear who is being referred to when Our Lord says ‘this rock’.
There is some contention about the genders of the words, saying that ‘petros’ means small rock and ‘petra’ means large rock. This in incorrect, as this distinction only exists in Attic (Classical) Greek, but the New Testament was written in Koine (Hellenistic) Greek which has no such distinction. 
Even if there were a distinction in the Greek words, it wouldn’t matter because Jesus and the Apostles spoke Aramaic. Greek was the lingua franca of the time, the language of business, writing, and science. But, most people had their own languages, and the Jews were not an exception. A few times in the epistles of St. Paul we read the Aramaic version of his name, ‘Cephas’ this is the transliteration of the Aramaic ‘Kepha”. So, what Christ would have said in Aramaic was, “But, I tell you, you are Kepha (Rock) and upon this kepha (rock) I will build my Church,”. This is the translation that St. Tatian records for us in the Diatresson 23:37. And this clearly shows the parallelism Christ is trying to establish between St. Peter being the rock and this rock being what Christ builds his Church on.
About this passage the Catholic Encyclopedia states, “He [St. Peter] is to be the principle of unity, of stability, and of increase. He is the principle of unity, since what is not joined to that foundation is no part of the Church; of stability, since it is the firmness of this foundation in virtue of which the Church remains unshaken by the storms which buffet her; of increase, since, if she grows, it is because new stones are laid on this foundation.”
It is also through St. Peter, says Christ, who will ultimately grant the Church victory over her ancient adversary, Satan. Christ is granting to St. Peter, authority over issues that are solely Christ’s. St. Peter is the foundation stone of the Church, yet so is Christ (Acts 4:11; Ephesians 2:19-22; 2 Peter 2:4-8); St. Peter is come to destroy the works of the devil, yet so is Christ (1 John 3:8). Christ is establishing St. Peter as the principle of unity, stability, and increase in the Church; yet Christ is all these things and more for his Church. Christ is “therefore assigning to Peter, of course in a secondary degree, a prerogative which is His own, and thereby associating the Apostle with Himself in an altogether singular manner.” 
“I will give you the keys of the kingdom of heaven, and whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven. (Matthew 16:19)” Keys are a symbol of authority, that is what is being granted to St. Peter here, authority over the Church. Then, the extent of this authority is shown, that whatever he binds is bound in heaven, and whatever he looses, is loosed in heaven. This is called the authority to bind and loose, and it is a rabbinical term.
The Jewish Encyclopedia says this, “The power of binding and loosing was always claimed by the Pharisees. Under Queen Alexandra, the Pharisees, says Josephus (“B J.” i, 5, § 2), ‘became the administrators of all public affairs so as to be empowered to banish and readmit whom they pleased, as well as to loose and to bind.’ This does not mean that, as the learned men, they merely decided what, according to the Law, was forbidden or allowed, but that they possessed and exercised the power of tying or untying a thing by the spell of their divine authority, just as they could, by the power vested in them, pronounce and revoke an anathema [curse] upon a person. The various schools had the power ‘to bind and to loose’; that is, to forbid and to permit (Ḥag. 3b); and they could bind any day by declaring it a fast-day (Meg. Ta’an. xxii.; Ta’an. 12a; Yer. Ned. i. 36c, d). This power and authority, vested in the rabbinical body of each age or in the Sanhedrin (see Authority), received its ratification and final sanction from the celestial court of justice (Sifra, Emor, ix.; Mak. 23b).” 
Here, the unique unification with prerogatives that belong to Jesus are present and move forward in this verse as well. St. Peter is granted keys and the authority to bind and loose, and Jesus already has these keys and the authority to bind and loose (Revelation 3:7). St. Peter is being granted the singular authority of Christ for the Church in Christ’s soon to come absence from the visible world.
There are three parallels to this event in other parts of Scripture, and we’re going to look at two of those:
“The proposal pleased Pharaoh and all his servants. Pharaoh said to his servants, ‘Can we find anyone else like this—one in whom is the spirit of God?’ So Pharaoh said to Joseph, ‘Since God has shown you all this, there is no one so discerning and wise as you. You shall be over my house, and all my people shall order themselves as you command; only with regard to the throne will I be greater than you.’ And Pharaoh said to Joseph, ‘See, I have set you over all the land of Egypt.’ Removing his signet ring from his hand, Pharaoh put it on Joseph’s hand; he arrayed him in garments of fine linen, and put a gold chain around his neck. He had him ride in the chariot of his second-in-command; and they cried out in front of him, ‘Bow the knee!’ Thus he set him over all the land of Egypt. Moreover Pharaoh said to Joseph, ‘I am Pharaoh, and without your consent no one shall lift up hand or foot in all the land of Egypt.’ Pharaoh gave Joseph the name Zaphenath-paneah; and he gave him Asenath daughter of Potiphera, priest of On, as his wife. Thus Joseph gained authority over the land of Egypt. (Genesis 41:37-45)”
Jesus says, “flesh and blood has not revealed this to you, but my Father in heaven”; Pharaoh said, “Can we find anyone else like this—one in whom is the spirit of God” and “Since God has shown you all this, there is no one so discerning and wise as you”. Jesus said, “I will give you the keys of the kingdom of heaven”; Pharaoh removed “his signet ring from his hand” and “put it on Joseph’s hand”. Jesus said, “whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven”; Pharaoh said, “all my people shall order themselves as you command” and “without your consent no one shall lift up hand or foot”.
As one can see, the parallels are striking between these two passages, and just as Pharaoh said, “only with regard to the throne will I be greater than you” so likewise, Christ is exalting St. Peter so high that only in regard to Christ’s deity and enthronement on heaven is he greater than St. Peter now for a look at the next parallel:
“On that day I will call my servant Eliakim son of Hilkiah, and will clothe him with your robe and bind your sash on him. I will commit your authority to his hand, and he shall be a father to the inhabitants of Jerusalem and to the house of Judah. I will place on his shoulder the key of the house of David; he shall open, and no one shall shut; he shall shut, and no one shall open. I will fasten him like a peg in a secure place, and he will become a throne of honor to his ancestral house. And they will hang on him the whole weight of his ancestral house, the offspring and issue, every small vessel, from the cups to all the flagons. (Isaiah 22:20-24)”
Jesus said, “And I tell you, you are Peter, and on this rock I will build my church”; and Isaiah prophesied, “I will fasten him like a peg in a secure place, and he will become a throne of honor to his ancestral house. And they will hang on him the whole weight of his ancestral house”. Jesus said, “I will give you the keys of the kingdom of heaven”; and Isaiah prophesied, “I will place on his shoulder the key of the house of David”. Christ said, “whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven”; and Isaiah prophesied, “he shall open, and no one shall shut; he shall shut, and no one shall open”.
One can see these parallels between these passages. And in the same way that Eliakim was made to be father to all of Israel (“the inhabitants of Jerusalem and to the house of Judah”) and hold an office with succession, as Eliakim is succeeding Sebna (Isaiah 22:15). In the same way was St. Peter made these things. We are not going to look at the last parallel because it just repeats information we have already discussed elsewhere. For the curious reader it is Esther 8:1-8.
“When they had finished breakfast, Jesus said to Simon Peter, ‘Simon son of John, do you love me more than these?’ He said to him, ‘Yes, Lord; you know that I love you.’ Jesus said to him, ‘Feed my lambs.’ A second time he said to him, ‘Simon son of John, do you love me?’ He said to him, ‘Yes, Lord; you know that I love you.’ Jesus said to him, ‘Tend my sheep.’ He said to him the third time, ‘Simon son of John, do you love me?’ Peter felt hurt because he said to him the third time, ‘Do you love me?’ And he said to him, ‘Lord, you know everything; you know that I love you.’ Jesus said to him, ‘Feed my sheep.’ (John 21:15-17)”
The Catholic Encyclopedia states, “The promise made by Christ in Matthew 16:16-19, received its fulfilment after the Resurrection in the scene described in John 21. Here the Lord, when about to leave the earth, places the whole flock — the sheep and the lambs alike — in the charge of the Apostle. The term employed in 21:16, ‘Be the shepherd [poimaine] of my sheep’ indicates that his task is not merely to feed but to rule. It is the same word as is used in Psalm 2:9 (Septuagint): ‘Thou shalt rule [poimaneis] them with a rod of iron’.” 
There is an interesting parallel between this ‘Petrine’ scripture and the other one in Matthew 16. In both, Jesus asks a question prompting an answer from St. Peter, and then blessing him with a singular unification of the prerogatives of Christ and St. Peter. Here, Christ establishes St. Peter as Shepherd of the entire Church, which Christ already is (John 10:1-18).
Some object to this interpretation and say that Christ in this passage is reinstating St. Peter to the Apostolic commission which he had lost by his three-fold denial. But, this doesn’t make sense because nowhere in Scripture does it make the claim that St. Peter lost his Apostolic commission, and when Christ was resurrected and gave the Apostles the authority to forgive sins (John 20:21-23) St. Peter gained the same authority as the others, so clearly St. Peter was not being reinstated into the Apostolic commission which he had never lost.
This is shown very well in the Council of Jerusalem in Acts 15. So, it starts off with a problem, “Then certain individuals came down from Judea and were teaching the brothers, ‘Unless you are circumcised according to the custom of Moses, you cannot be saved.’ (vs. 1)” This caused contention with Sts. Paul and Barnabas, “And after Paul and Barnabas had no small dissension and debate with them, Paul and Barnabas and some of the others were appointed to go up to Jerusalem to discuss this question with the apostles and the elders. (vs. 2)” The missionary apostles , Sts. Paul and Barnabas went down to Jerusalem to see what this was all about, if this was really what the Church was teaching.
“When they came to Jerusalem, they were welcomed by the church and the apostles and the elders, and they reported all that God had done with them. But some believers who belonged to the sect of the Pharisees stood up and said, ‘It is necessary for them to be circumcised and ordered to keep the law of Moses.’ The apostles and the elders met together to consider this matter. (vs. 4-6)” So they initiated a council on this matter because it was causing division in the Church. Then in verse 7 it says, “After there had been much debate, Peter stood up and said to them…” And he goes to give his teaching until verse 11. After St. Peter gives his teaching, the Greek word is “Ἐσίγησεν [Heh-sig-ais-ehn]” which in an aorist indicative which means that everyone became silent and stayed silent. After St. Peter gives his teaching, all debate is finished, the matter is settled.
Afterwards, Sts. Peter and Barnabas support St. Peter’s proclamation by sharing miraculous occurrences. And then St. James gives a speech to his Jewish brethren at Jerusalem, where he ties into what St. Peter has proclaimed, “My brothers, …Simeon has related …we should not trouble those Gentiles. (vs. 13-21)” as who were the ones troubling the Gentiles in the first place? James’ disciples. One can clearly see St. Peter’s primacy over the Church in this council, as well as his infallibility.
“Your sister church in Babylon, chosen together with you, sends you greetings; and so does my son Mark. (1 Peter 5:13)” This statement from St. Peter is a code word for his dwelling in Rome. Babylon is used frequently in contemporary texts to mean Rome. Just like biblical authors renamed Jerusalem: Zion and Ariel (Isaiah 29), and modern authors named New York: Gotham (Washington Irving, Salmagundi (1807)), so many authors around this time used Babylon to refer to Rome, such St. John in the book of Revelation, the Sibylline Oracles, the Apocalypse of Baruch, and 4 Esdras. 
So, what do all these quotes and studies of the Sacred Scriptures tell us about the papacy? They tell us that the Pope has ultimate teaching authority (Genesis 41:38-39; Matthew 16:17) as well as ultimate disciplinary authority (Genesis 41:40-44; John 21:15-17) and that his authority has primacy over the other bishops (Genesis 41:40, 45; Isaiah 22:15, 21-22; Esther 8:8; ). It also teaches us the historic reality that Christ established his Church on St. Peter (Matthew 16:18) and that he gave to St. Peter and his successors the keys to the kingdom (Genesis 41:42-43; Isaiah 22:22;Esther 8:2; Matthew 16:19), and that St. Peter has successors (Isaiah 22:20-21). It also teaches that salvation is impossible outside of communion with the Pope (Matthew 16:18; John 21:15-17). Also, papal infallibility (Acts 15:7-11) and his association with Rome (1 Peter 5:13) are also mentioned. This is the summary of what the Sacred Scriptures teach about one of the oldest doctrines of the Church, that of the Papacy.
Part I, where I go over the official Church teaching.
Part III, where I go over the teaching of the Church Fathers.
All scripture citations taken from the NRSVCE, the official liturgical scripture sanctioned by the Canadian Conference of Catholic Bishops (CCCB).
Addendum: In all things I submit to the authority of the Church if I have misrepresented any aspect of doctrine or history.