I tell you, you are Peter, and on this rock I will build My church, and the gates of hell shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven." This was in response to Peter's profession that Jesus is "the Christ, the Son of the living God" in verse 16. In Greek, the name Peter means "a rock," so Rome has long held that Jesus was naming Peter the foundation of the church, and the Pope as Peter's successor.
I have addressed this question before, but I am going to take a different tack here.
One thing to note is that these particular words of Christ don't appear in the parallel passages in Mark 8:27-30 and Luke 9:18-22. Both of those gospel writers stopped after Peter's profession, leaving out the response of Jesus. If Jesus had been making such a fundamental declaration, wouldn't it have been included in the parallel accounts?
Moreover, in all three Synoptics, an account is given shortly after this one, in which the apostles were arguing over leadership. See Matthew 18:1-4, Mark 9:33-37, and Luke 9:46-48. Notice especially the words of Jesus recorded in Mark 9:35: "If anyone would be first, he must be last of all and servant of all."
These three passages tell us two important things regarding the matter at issue here between Rome and Protestants (and the Eastern Orthodox Churches, as well). The first is that the other apostles didn't see the words of Jesus as appointing Peter to supremacy within their group. And second, they tell us that leadership in the church is not a matter of Jesus's call to supremacy, but rather His call to service. The palaces, gold trappings, and expensive art of the Vatican speak much of the former, but nothing of the latter.
There is, however, an account in Scripture regarding someone's claiming supremacy in the Church (III John 1:9-11): "I have written something to the church, but Diotrophes, who likes to place himself first, does not acknowledge our authority." The person who claims supremacy is an enemy of the church, not her head.
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