"he katholike ekklesia (the universal Church) was an expression first documented by St. Ignatius, Bishop of Antioch,"
Then there is no scriptural basis for calling the Antioch Church "Catholic" nor is there a scriptural basis for it being founded by Peter.
I didn't realize that all Church history ended at Acts 28. Silly me.
This is where one leaves Acts and reads extant literature for an understanding of the early Church.
"See that ye all follow the bishop, even as Christ Jesus does the Father, and the presbytery as ye would the apostles. Do ye also reverence the deacons, as those that carry out the appointment of God. Let no man do anything connected with the Church without the bishop. Let that be deemed a proper Eucharist, which is [administered] either by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear, there let the multitude also be; by the bishop, or by one to whom he has entrusted it. Wherever the bishop shall appear, there let the multitude also be; even as, wherever Jesus Christ is, there is the Catholic Church." Ignatius of Antioch, Epistle to the Smyrneans, 8:2 (c. A.D. 110).
"[A]ll the people wondered that there should be such a difference between the unbelievers and the elect, of whom this most admirable Polycarp was one, having in our own times been an apostolic and prophetic teacher, and bishop of the Catholic Church which is in Smyrna. For every word that went out of his mouth either has been or shall yet be accomplished." Martyrdom of Polycarp, 16:2 (A.D. 155).