Would an NP-complete public key cryptosystem imply NP=co-NP?
Your argument is correct about the FACTORING problem, but it does not automatically generalize to all public-key cryptosystems. 

Here's an excerpt from an exercise by Dominique Unruh, showing formally that if FACTORING is NP-complete, then NP = coNP.







If you are interested in research on the relationship between crypto and NP-completeness, I suggest reading Pass's paper Parallel repetition of zero-knowledge proofs and the possibility of basing cryptography on NP-hardness.