The term "Roman Catholic" was invented by Anglican enemies of Rome. Not surprisingly, strictly speaking, nobody considers himself a Roman Catholic (except, perhaps Catholics who actually live in the diocese of Rome). It really should be much of a surprise that nobody living in the first century would call himself by an appellation that would not be invented for 1500 years.
However, in AD 110, Ignatius of Antioch referred to himself as "a bishop of the Catholic Church". Ignatius knew some of the Apostles personally, and was third in line in the see of Antioch, after Peter.
nor did the Roman Catholic Church as it is today exist at that time.
Nor, of course, did anything else "as it is today exist at that time". That should not come as a surprise to anyone.