"White" is a very confusing term which should just be avoided in these discussions. Even talking purely about ancestry of ancient peoples (including Romans), it conflates very diverse origins - Old European farmers (reflected today most clearly by Sardinians), Near Eastern/Levantine (found peculiarly in ancient Etruscans, which agrees with their mythology that tells of a descent from some ancient people of Lydia), Eurasian steppe peoples (this would be the PIE culture-related component that people tend to think about most readily, but there was a lot less of it in the urbanized Mediterranean area than farther north in continental Europe!) were quite distinguishable back then, even though all of them might be described very loosely as "white".