This conclusion is confirmed by the unanimous voice of tradition which, as early as the second half of the second century, designates the Prince of the Apostles the founder of the Roman Church. (St. Peter Prince of the Apostles, New Advent)
The Profession of faith of the other fathers added: and I pledge and swear true obedience to the Roman pontiff, successor of blessed Peter the prince of the apostles, and vicar of Jesus Christ. (1st Vatican Council, Session 4)
Peter Never Made a Mission Trip to Rome
This might be a good time to state that Peter was never sent to Rome on a mission, nor did he claim to travel there. Peter had made mission trips to Joppa (Acts 10:31), Samaria (Acts 8:14), Caeserea (Acts 10:23), Jerusalem, and possibly Judea (Acts 1).
Rome is explicitly not included there, so it is not scripturally or historically sound for Peter to be the founding apostle and lead bishop of Rome.
The Scriptures State Paul Founded the Church in Rome
Contrary to Roman Catholic tradition, Paul actually infers himself to be laying the foundation of Christiandom in Rome in his letter to the Romans:
Romans 15:20: And in this way I desire to preach where Christ has not been named, so as not to build on another person’s foundation.
He opens the epistle by expressing his intention to minister to the Romans, and closes it by explaining that he chooses not to minister in regions where Jesus is already by being preached. With a bit of deductive reasoning, it’s apparent that there wasn’t an legitimate church in Rome prior to Paul’s arrival. Furthermore, when Paul is later brought to Rome for trial, he writes this to Timothy:
2 Timothy 4:16: At my first defense no one appeared in my support; instead they all deserted me—may they not be held accountable for it.
Paul being the founder of the Christian church in Rome is not just scripturally sound, but logically supported, too. Why would Paul claim to have singled out Rome for never having received the Gospel of Christ if Peter already resided there? Why would Paul later claim to be alone when he was on trial in Rome if Peter had become the head Bishop there?
Not only that, but think back to my explanation of the Galilean wedding for the metaphorical basis for Jesus’s promise during the Last Supper. Why did the command to have communion in remembrance of Jesus persist in the Early Roman Catholic church throughout the centuries when the context of the Galilean wedding didn’t?
Perhaps it was because Paul, likely the only non-Galilean apostle, was the one to bring the Christian church to Rome, where Roman Catholicism originated.
The references Jesus had made to Galilean marriage tradition would not have had the same weight to Paul, who would not have been as immersed in that culture.
What About Babylon?
1 Peter 5:13: She who is at Babylon, who is likewise chosen, sends you greetings, and so does Mark, my son.
A common claim is Peter referred to Rome as Babylon, though it is nowhere mentioned in scripture. No other apostle refers to Rome as Babylon, either, complicating this claim. But, Peter using Babylon as a codename for Rome makes a little more sense than Peter actually preaching and writing from Babylon, since the book of 1st Peter was written late in his life and he historically dies in Rome.
So, if we accept this claim at face value, does this mean that Peter did found the church in Rome and was its first and primary Bishop? No, not even remotely. 1 Peter was written in the early-mid 60s A.D., the book of Romans was written in the late 50s A.D.
Sources: Paul, Apostle of Christ, Chronological Order of the Letters, Who Was Paul?, The Letter to the Romans – Bible Gateway and When was the letter of 1 Peter written? – Evidence for Christianity
With the proper timeline in context, we know that Paul founded the church in Rome. We also know scripturally that Peter had no power, (religious or political) in Rome at the time to support Paul during his trials there. That pretty much dashes any claim that Peter presided in Rome as its bishop for decades, too.
In Conclusion
So, the assertion that Peter established a supernatural line of bishops, which he himself never claimed to have done, in Rome, where he never claimed to have built a church, is more than a little ludicrous.
If we judge it by history, the only time Peter may have ever been to Rome was leading up to his execution under Nero’s reign, which was recorded as being roughly around the same year he wrote his epistles (64-65 A.D.).
Source: How Did Peter Die – Was He Crucified Upside-down? (crosswalk.com)