It is St. Peter that is the key, not specifically Rome, as Rome was his
final stop in the history of his ministry:
"For roughly its first decade of existence, the Church was made up exclusively of Jews in the area of Jerusalem. But as the Church grew and spread across the Roman Empire, it incorporated Jews and Gentiles, rich and poor, Romans, freemen, and even slaves—men and women from every tribe and tongue. By the third century, one out of ten people in the Roman Empire was a Catholic. Just as the word
Trinity was appropriated to describe the nature of God, so the term
catholic was appropriated to describe the nature of Christ’s mystical body, the Church.
But let’s get back to the history of the word
catholic. The first recorded use of the word is found in the writings of Ignatius of Antioch, who was a young man during the time of the apostles and was the second bishop of Antioch following Peter, who went on to become bishop of Rome. Ignatius was immersed in the living traditions of the local church in Antioch, where the believers in Christ were first called Christians (cf. Acts 11:26). He was taught and ordained directly by the apostles. From the apostles Ignatius learned what the Church was—how it was to function, grow, and be governed.
History informs us that Peter was the bishop of Antioch at the time; in fact, Church Fathers claim that Ignatius was ordained by Peter himself. Ignatius must have worshiped with Peter and Paul and John. He lived with or near them and was an understudy of these special apostles. Ignatius is known and revered as an authentic witness to the traditions and practice of the apostles.
In the existing documents that have come down to us, Ignatius is the first to use the word
catholic in reference to the Church. On his way to Rome, under military escort to the Coliseum, where he would be devoured by lions for his faith, he wrote, “You must all follow the bishop as Jesus Christ follows the Father, and the presbytery as you would the apostles.
Wherever the bishop appears, let the people be there; just as wherever Jesus Christ is, there is the Catholic Church" (
Epistle to the Smyrnaeans 8).”
catholic.com/thisrock/2005/0501fea4.asp
In the final quote above in bold, you can see that the Bishop of
Antioch affirms that the Catholic Church is the True Church.