How would you handle a situation where another team member was critical of your work?

The first thing I would do is listen to that member’s critique of my work. Then I would assess the correctness of the critique - likely by inviting others familiar with the subject to assess both my work and the critique.

Based on this analysis I would either
A) accept the critique as accurate and thank the person offering it for correcting my error, or
B) offer the counter-argument showing their critique to be in error and requesting a withdrawal of that critique. I would do this privately and, if that did not succeed, repeat the request at a team meeting after reviewing all the information on both sides. If that fails, take it to your boss or, if necessary, his boss.

The important thing is to settle it as quickly as possible since your reputation is being questioned.