The doctrine of the papacy is unique to the Catholic religion. When reading the Bible, it's not hard to see why. The Bible paints a very different picture of Peter than the Catholic Church does.
Let us start with Matthew 16:18. This is the verse that Catholics appeal to to establish Peter's primacy. Is Peter the rock upon which the Church is built? This is a sticky issue. While the First Vatican Council dogmatically stated that the Church has always understood this verse to mean Peter is the first Pope, this is demonstrably false. There are actually five views that have been taken throughout history. In ascending order of popularity, these are:
- Peter is the Pope
- All the Apostles were the rock.
- Jesus Himself is the rock.
- Peter is the rock.
- Peter's confession that "You are the Christ, the Son of the living God" is the rock.
Catholics insist that Jesus must have been referring to Peter, because Peter means rock. In the original Greek, however, the word used for Peter is different from the word used for rock. Jesus said "you are Petros, and on this petra I build my Church." Some Catholics counter this by saying that actually, Jesus was speaking Aramaic, so it would have actually been "you are Kephas, and on this kephas I build my Church". The first problem with this is that we're speaking entirely hypothetically at this point. Aside from the fact we have no Aramaic copies of Matthew's Gospel, there are still more ways Jesus could have differentiated between Peter and the rock in Aramaic as well, so really, the Catholic argument is just wishful thinking.
What we do have is the Greek, which brings us to the second problem: The Greek is inspired. What that means is that even if Jesus did explicitly say Kephas both times, the Holy Spirit guided Matthew to write Petros and petra, which He would only have done if Jesus intended to separate the rock from Peter. Of course, on the flip side, if Peter really was the Pope, and this verse was intended to prove it, then the Holy Spirit, when He inspired it, would have foreseen the confusion caused by Matthew's alleged translation error and inspired him to correct it. The context of the verse as a whole ultimately suggests the dominant historical view (that Peter's testimony is the rock) is correct.
But whether or not Peter is the rock is ultimately irrelevant. Rock and Pope are not the same thing. The word "Pope" originally meant father, and it has changed hands repeatedly throughout history. Even the Catholic Encyclopedia admits "As regards the title Pontifex Maximus, especially in its application to the pope, there was further a reminiscence of the dignity attached to that title in pagan Rome".
There are multiple other ways in which Peter could have been called a rock, or even the rock, which do not require him to have been some kind of super-Apostle. He was the first to proclaim the Gospel on the Day of Pentacost (Acts 2:14-47), he was the first to take the Gospel to the gentiles (Acts 10:1-48), he seeded several early congregations, and he wrote scriptures. Ephesians 2:20 tells us that the Church is built on the foundation of the prophets and Apostles. Peter was, of course, a believer, which makes him a living stone (1 Peter 2:4). Peter didn't have to be a Pope in order to be a rock on which the Church is built in any of these senses. Therefore, even if you impose the idea that Peter is the rock onto Matthew 16:18, which is extremely hard to do, it doesn't even come close to supporting the Papacy.
This is further backed up by the fact that Jesus put the other Apostles on par with Peter when He gave them the same binding and loosing authority in Matthew 18:18. In Catholicism, there is only one Pope, and all 11 other Apostles are seen as inferior to Peter whether taken alone or together. Yet there is no evidence that Peter had any authority that the other Apostles, even Paul, had. In fact, though Paul considered himself unworthy to be an Apostle, he actually declared himself to be behind the most eminent Apostle in nothing (2 Corinthians 11:5; 2 Corinthians 12:11).
The other Apostles knew Matthew 16:18 wasn't bestowing Peter primacy, too. After Jesus had declared "on this rock I build my Church", a dispute arose among the Apostles as to which of them was the greatest (Luke 22:24). Imagine cardinals arguing with each other over whether they might possibly be greater than the Pope! It couldn't realistically happen. Yet, the Apostles did have this dispute. Jesus, of course, said "you bunch of dumplings! I've already told you Peter's greater than the lot of you put together." Sarcasm aside, Jesus didn't even hint that Peter was preferable to any of them, but rather He said let the greatest of them be the one who serves. This means that, theoretically speaking, Peter could have gained the title of the greatest Apostle if he was humble enough to serve his way to greatness. But even if we find evidence that he was humble enough to earn this honour, he didn't get it simply by being chosen.
There is some evidence that Peter was humble enough to serve his brethren at least sometimes. In Acts 8:14, they sent him, and he obeyed. It doesn't seem especially papal to obey the commands of a lesser authority. But the real clincher comes from Galatians 2, when Paul, the chief of sinners, stood up to Peter on a matter of faith. Now, to be fair to Catholics, papal infallibility only applies when the Pope speaks "ex cathedra", which Peter wasn't doing (mostly because this doctrine hadn't evolved yet...), but nevertheless, one can hardly imagine an ex-persecutor of the Catholic Church converting, becoming a Cardinal, walking up to the Pope, and telling him, to his face, that his philosophy means Christ died in vain.
So, Biblically, the papacy just doesn't exist. Peter doesn't resemble a Pope in any way, shape or form. Neither did the earliest Popes, as the papacy is an ever-evolving doctrine that was initially stolen from Roman paganism. I simply cannot understand how so many people believe the Pope is the head of the Church.