The New Testament was written by individual apostles, and their disciples, Catholic clergy all.
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Uhhhh, no.
The Roman bureaucracy took root, what, 300-400 years post Peter.
The whole Peter thing is more than a little arguable.
Calling anyone penning Scripture Catholic in anything remotely close to the Roman sense strikes more than a few of us as gross usurpation and cheekiness.
Only because you arbitrarily place the beginning of the Catholic Church at the Decree of Theodosius, while we consider the Pentecost to be the moment the Chruch was born. The term was first used by St. Ignatius in 110 AD, in the same sense we use it today.