From the article on the Wikipedia “Pope”, it was stated that the Pope is the successor of St. Peter as bishop of Rome.Even though Peter was the Apostle to the Jews, he can still be considered the Bishop of Rome.
The Pope (from Greek: pappas, father; from Latin: papa, Papa, father1) is the head of the Catholic Church. He is the Successor of St. Peter as the Bishop of Rome. (link)
It made me wonder if the St. Peter mentioned here is Apostle Peter, because the Peter in the Bible who was an Apostle of Christ was not assigned to be the Apostle in Rome but in Jews. In Galatians 2:8, Apostle Paul clearly stated:
8 For God, who was at work in the ministry of Peter as an apostle to the Jews, was also at work in my ministry as an apostle to the Gentiles. (NIV)
The Apostle who was assigned to the Gentiles, including Rome, was not Peter but Apostle Paul.
If Peter was assigned to be the Apostle to the Jews and Apostle Paul as the Apostle to the Gentile, how could it be that the Roman Catholic Church claimed that Peter was the first Bishop of Rome? I think, it is more believable if they will say that Apostle Paul is the first Bishop of Rome because it was he who was assigned to the Gentile nations including Rome.
First of all, it's helpful to note that, even though Peter's evangelistic efforts were perhaps directed primarily to the Jews, his ministry was not exclusively to them. After all, he was the one who spoke with the Spirit at Pentecost to men of all nations (cf. Acts 2:14-41). Also, it was to Peter that God gave the vision validating the Gentiles as men worthy of the Gospel (cf. Acts 10:9-16,28; 11:5-10), and it was Peter who converted the household of Cornelius, the first Gentile converts (cf. Acts 10:44-48; 11:15-18). Finally, Peter himself says, at the Council of Jersualem: "Brothers, you know that some time ago God made a choice among you that the Gentiles might hear from my lips the message of the gospel and believe" (Acts 15:7).
So, Peter's ministry was certainly for the Gentiles too.
Secondly, Peter is considered the bishop of Rome, because he, the pre-eminent apostle, lived there (at least for a short time) and was martyred there. This is a historical fact attested to by many early documents (see this link).
Ultimately, Peter is the Pope because he is the Rock upon whom Jesus founded his Church and he was the one given the keys of the kingdom of heaven (cf. Mt 16:16-20). Jesus prayed for Peter, that his faith would not fail (cf. Lk 22:32) and it was to Peter that Jesus gave the mission to feed and tend the sheep, including the other apostles (cf. Jn 21:15-19).
I hope that answers your question.