It is an official Catholic dogma that when Jesus said, in Matthew 16:18, "you are Peter, and on this rock I build my Church", Jesus was bestowing the papacy upon Peter. First Vatican Council even declared that anyone who does not believe this is anathema. That being the case, Catholics simply cannot afford to let this interpretation go, and so the amount of time and effort Catholics put into defending it is astounding.
One of the "better" arguments used to defend this interpretation is that in verse 19, Jesus says "And 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." Catholics argue that this proves Peter was the first Pope, because in Isaiah 22, the key of David is promised to Eliakim, granting him control of David's domain.
Isaiah 22:22 says "The key of the house of David I will lay on his shoulder; So he shall open, and no one shall shut; And he shall shut, and no one shall open." One thing that should be immediately obvious is that there is a major difference between the key (singular) of David (the king of Israel) and the keys (plural) of the Kingdom of Heaven (God's Holy Kingdom). There is a reason for this: They're not the same keys.
In Matthew 16, Jesus continues to clarify exactly what the keys of the Kingdom of Heaven do: "...whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven." I've addressed the meaning of this phrase before. But what's even more interesting to this discussion is that in Matthew 18:18, Jesus gave the same authority to all the Apostles. This is likely why many people throughout history, including Augustine, believed all of the Apostles were the rock described in Matthew 16:18. But how many Popes are there? Theoretically, only one Pope can reign at once (though the Western Schism is a notable exception). There are never supposed to be 12 simultaneous Popes tying their claims to authority to individual Apostles.
But if Peter doesn't hold the key of David, who does? Revelation 3:7 gives us the answer: "And to the angel of the church in Philadelphia write, ‘These things says He who is holy, He who is true, “He who has the key of David, He who opens and no one shuts, and shuts and no one opens”:" This is a verbatim quote of Isaiah 22:22, and so whoever it is saying these things to the Church in Philadelphia, He is the one holding the key of David. This is Jesus! Jesus holds the key of David.
In no less than three places in scripture (Ephesians 1:22; 5:23; Colossians 1:18), Jesus is explicitly called the head of the Church. Peter receives that title zero times. If Catholics are willing to give Peter a title that rightfully belongs to Jesus, one more blasphemy such as claiming Peter holds the key of David which actually belongs to Jesus is unsurprising. The Catholic Church seems to love taking that which rightfully belongs to God and giving it instead to some inferior being.
As I have just demonstrated, pretty much the most powerful argument Catholics rely on to prove a Biblical basis of the papacy is a lie. And so the question to ask is this: If Catholicism is telling such a huge lie to defend its leader, where is he leading Catholics to?