I'm not sure of that. Christianity, like Judaism, was very varied during this period. In both religions, people developed - over quite a long period of time - the notion that "correctness" in both belief and practice was a Thing in the first place, and attempted to define what it was. So there are parallels between them, but I don't know of any reason to think that rabbinic Judaism was influenced by Christianity on this score. Rather, the major impetus behind its rise and the transformation of Judaism was the loss of the Temple, which had previously been the focal point of the whole religion.
This sort of thing happens all the time. A similar example might be Zhu Xi's systematisation and synthesis of Confucianism, in which he defined the canonical texts and set out what he thought were the central doctrines. That certainly didn't owe anything to Christianity, but it's a natural development of any system of thought that's been around for a long time and has developed all kinds of variants.