Initially I had the same concern, but after digging a bit deeper, I don't believe this would be an immediate outcome. From the little I can gather, the ABC conjecture is really just a statement about the relationship between the operations of addition and multiplication on triples of coprime positive integers. While prime factors are involved in the conjecture, it doesn't seem that the proof of the conjecture would naturally lead to prime factorization in polynomial time. I'm not a mathematician though, and given how radical Mochizuki's proof sounds, I imagine it may (if it stands) eventually lead to more efficient factorization algorithms.

Advances in computing power seem to be a more worrisome threat to RSA, especially for RSA-1024 whose factorization is probably already feasible (http://www.cs.tau.ac.il/~tromer/papers/cbtwirl.pdf). And once advances in quantum computing become reality, quantum algorithms should be able to very quickly break even RSA-2048 (http://en.wikipedia.org/wiki/Shors_algorithm).