The first passage ReformedApologist brings up is 1 Peter 5:1-2, in which Peter calls himself a "fellow elder," which, according to my opponent, shows that Peter was equal to all the other bishops of the Church. The problem with this interpretation, though, is that it proves too much. If we take this reasoning to its logical conclusion, we would conclude that Peter was not even an apostle, which we know is false. Thus, Peter could not have meant what he wrote to be taken hyper-literally. So, what did he mean? As a bishop, Peter WAS a fellow elder, just as we Americans are the president's "fellow Americans." However, we're not all presidents, just as not all elders are popes.
Next, he brings up the Council of Jerusalem and says that nothing indicates that Peter acted as pope. However, I already dealt with this in my opening statement. Peter made the infallible decision (verses 7-11), and all who spoke afterwards were merely supporting him. Then, James suggested a certain pastoral implementation of the doctrine that the rest of the council agreed to. There is absolutely nothing in this passage that goes against Petrine primacy. Remember, just because Peter presided over the council as pope and had the final say doesn't mean that the other apostles and elders didn't play important roles in decision-making as well.
The third passage he brings up is Galatians 2:6-9, in which Paul says, "what they were makes no difference to me; God shows no partiality," and ReformedApologist holds that Paul was referring to authority. Thus, he concludes, Peter and Paul were equals. The problem, however, is that if taken to its logical conclusion, this would prove that all Christians have equal authority (or at least that the elders and apostles had equal authority), but we know that the apostles had more authority than anyone else. So, what could Paul have meant? I believe Paul was saying that God shows no partiality in His mercy (which fits more naturally with the KJV and Douay-Rheims translations).
The fourth passage ReformedApologist brings up is 2 Corinthians 12:11, which he takes out of context. Read in context (2 Corinthians 11:4-5, 13), we can see that Paul is referring to false apostles and sarcastically calling them "super-apostles." Then, ReformedApologist mentions the fact that Paul was the apostle to the Gentiles, while Peter was the apostle to the Jews. However, an apostle is one who is sent to preach, so whether Peter was sent to preach to all or some people has nothing to do with his authority. Paul knew Gentile culture well, so he was the natural choice for evangelizing them, but this has no bearing on whether or not Peter was the head of the Church.
Next, ReformedApologist brings up Galatians 2:11-14, which, as I said in my opening statement, does not refer to an official teaching of Peter; rather, it was a personal practice. Granted, he wasn't being straightforward about the Gospel, but it was through his practice, not his official teachings.
The sixth passage he brings up is 1 Corinthians 3:6-11, which has nothing to do with Peter. Paul says that he and Apollos are one, but he doesn't mention Peter, so we can't draw any conclusions about him based on this passage.
Next, ReformedApologist brings up 1 Peter 5:13, in which Peter says he is in Babylon. Many scholars believe that Babylon was a code word for Rome, and there is an abundance of evidence from the Church Fathers that Peter was in Rome. Besides, this passage is irrelevant for two reasons: 1) Peter very well could've been in Rome after or before he wrote this epistle, and 2) Even if Peter never was in Rome, that doesn't diminish his authority. His successors could've just (for whatever reason) moved his office to Rome after his death (papal authority is in the office, not the place where the pope resides).
Next, he discusses Matthew 16:17-19. He challenges any Catholic apologist to provide an extant Aramaic manuscript, but, as he knows, there are none. In addition, he gives two reasons based on the Greek text to reject the theory. I'm no scholar, so I can't debate this, but it's not that relevant. I didn't use the theory in my opening statement. However, even though we don't know for sure in what language the first Gospel was written, we do know that the Greek of Matthew is a translation of the Aramaic that Jesus spoke, and He most likely used the word "kepha" both times. Next, ReformedApologist just claims that because Matthew uses two words for "rock," Jesus was talking about two different things, but that makes the wording awkward. In my opening statement, I gave four reasons why Jesus was referring to Peter both times, and he will have to deal with those before he can claim that Jesus was referring to Peter's confession of faith.
Next, he says that in Matthew 18, all the apostles are given the keys of the kingdom of heaven, but that's not true. They had the power to bind and loose, which is part of the authority symbolized by the keys but is not synonymous with the keys, which only Peter received.
Then, ReformedApologist claims that Jesus, not Peter, fulfills Isaiah 22:22. Granted, Jesus ultimately has all the power, but He is the king, not the prime minister, so there must be someone below Him who fills that role. Therefore, when Jesus gave Peter the keys and the power to bind and loose, He installed him as the prime minister in the Church.
ReformedApologist ends with some quotes from the early Church Fathers and councils, which he believes demonstrate that the papacy is unhistorical. However, they fail to prove his point.
The Seventh Council of Carthage was a local, not ecumenical council that only dealt with African concerns. Elsewhere, St. Cyprian (who lead the council) attests to the primacy of Peter and his successors (The Unity of the Church 4; Epistle 59, 14).
Steve Ray, in his book Upon This Rock, gives us some insight into the quote from the 6th cannon of the Council of Nicaea:
This phrase can be understood in two ways. First, it is the custom of Rome to have jurisdiction over the bishops of surrounding territories, providing a justification for Alexandria to do so as well. Second, it is the custom of Rome to acknowledge the prerogatives of the Alexandrian bishop, giving the council the precedent for doing so. Either way, it shows Rome as the primary model and the basis for the determination of the other Churches.
In addition, the Arabic version of the canons of the Council of Nicaea says,
He who occupies the chair of Rome, is the head and prince of all patriarchs; since he is the first, as was Peter, to whom power is given over all Christian princes, and over all their peoples, as he who is the Vicar of Christ our Lord, over all peoples and over the whole Christian Church, and whosoever shall gainsay this is excommunicated by the Synod.
It is also interesting to note that while the council strengthened several patriarchs, it did not legislate any authority over Rome.
The quote from Pope Gregory the great is misleading. He condemns anyone who calls himself the "Universal priest," and ReformedApologist wants us to believe that he meant chief priest, but that's not what he intended. Later in that same epistle (actually in the part that ReformedApologist quotes from!), he says that this same person "covets being called sole priest," indicating that that's what the title "Universal priest" means; it has nothing to do with authority.
The last Church Father that ReformedApologist quotes is Cyprian who, as I said before, accepted the primacy of Peter and his successors. So, what did he mean in this last quote? He meant that Peter did not invoke his authority and rebuke Paul for rebuking him; rather, he humbly accepted the correction. In fact, if Peter was not the head of the Church, why would Cyprian even mention that Peter did not say that he held the primacy? It would have made no sense for Peter to have even thought of saying that. The only satisfactory answer is that Peter was predominant over the other apostles but didn't invoke his authority in that instance.