Please correct me if I am wrong, but I believe this question was prompted by the question on the main site entitled Why is Stouffer's method often used with one-tailed p-values rather than two-tailed p-values?
I strongly encourage you not to delete your answer!
I have dithered for at least a couple of days regarding whether to post an answer here, partly because I initiated the comment stream at that question.
I have decided I would like to add my perspective and a couple of other thoughts. I hope nobody minds.
I think that in this situation there are a number of viable options, and these options follow through in the general case as well.
Leave the answer as is. If you feel you have answered the question correctly and clearly and there is nothing more to add, then, by all means you have no obligation to make any changes to reflect any of the comments and, perhaps, you shouldn't. This may be particularly true if only a single person is raising objections that you feel are unfounded. In part, I believe this is one reason the voting mechanism is present. It doesn't work perfectly, but to first order, it is reasonably useful.
Make edits to clarify your answer and reenforce your assertions. If you feel you've answered the question correctly, but that, perhaps, part of the content is being inadvertently misinterpreted by others, then there may be ways to reword or restate things that will make what is clear to you more clear to others. This happens very frequently on this site and, at least in my opinion, generally leads to more complete and useful answers. Oftentimes, speaking solely from my own experience, I learn quite a bit from comments and the subsequent revisions to the question or answer that they generate. Finally, edits will "bump" the question back to the top of the queue, leading more users to view it and, hopefully, vote according to how the content has been revised.
Engage other users in chat or through another medium. Comments have an upper bound of 500 characters. This forces them to be unnecessarily terse sometimes, especially if $\LaTeX$ or a link is included. We have generally not made good use of the "chat" mechanism on this site. For the most part it is unneeded in day-to-day situations and a lot of our discussions are facilitated by $\LaTeX$ which isn't natively supported there. However, if a comment stream is becoming extended, it can probably be useful and a persistent transcript can then be linked back to the answer. If that fails, as a last resort, email communication could be useful. But, my feeling is that that's generally undesirable except in the most extreme of circumstances.
Delete your answer. My feeling is this should be absolutely the last resort and should only occur if you believe (a) your answer is obviously incorrect and (b) there is no reasonable hope of editing it in order to salvage it. This can and does occur, but it's rare and mostly happens when either (a) an answerer misreads a question and answers a different question instead or (b) the question gets edited in a way that completely changes its meaning, invalidating an answer in the process.
As noted recently, answers are the first-class citizens here and comments are rather looked down on by the site administrators. In fact, I believe answers are meant to be persistent and comments explicitly not. So, it's quite possible at some point that entire sets of comments could be wiped from the site, leaving only the answer.
A (not so) brief aside on my own usage:
I have a strong preference for utilizing comments on this site. This is mostly because I enjoy the engagement with other users and find that my own learning is more active that way. Occasionally, albeit rarely, the comments I post may be of some use to others as well. (I am also excruciatingly slow at formulating full-blown answers, which makes commenting a more efficient way for me to participate, generally speaking.) My opinion is it's generally better to have one very good answer than two very similar answers of less quality; commenting often achieves that goal. Others are likely not to agree with that philosophy.
The moderators have managed the tenor of this site perfectly in my view and besides my interest in the content, this is one reason I spend more time here than other sites in the SE network that tend to have a little more "edge" to the communication. Unfortunately, the character limit on comments sometimes forces them to be unnecessarily terse. This makes the tone a bit more difficult to project. If the tone of my comments—or those of others—ever sounds off, I would encourage everyone to give the commenter the benefit of the doubt and seek clarification where appropriate.