A method to efficiently factor large numbers will also break the ECDSA.

No, ECDSA relies on the hardness of the discrete logarithm problem. Nothing to do with factoring, at least not in the classical sense.

On a quantum computer, my understanding is that Shor's algorithm could potentially target both problems, though.

	▲	cyberax 6 days ago \| parent [-]
		Both systems are an example of a hidden Abelian subgroup problem. That is also why Shor's algorithm equally applies to both: https://en.m.wikipedia.org/wiki/Shor%27s_algorithm#Shor's_al... So a hypothetical classic algorithm that breaks the RSA is also highly likely to break the ECDSA.