I see basically three possible responses:
(a) Write a rebuttal ranting that this person is a jerk and his criticisms are totally unfair and that he isn't worthy to be spit upon by a great author like yourself, etc. Let's say we rule that out.
(b) Do nothing. Take your lumps.
(c) Write a rebuttal pointing out factual errors, but keep it as reasoned and non-argumentative as possible.
Perhaps I should say that in my case, I wrote a non-fiction book on a somewhat controversial subject, and someone with different beliefs posted a review on Amazon attacking my book that included assertions I consider inaccurate. (I'm thinking that I don't want to be more specific than that to avoid biasing answers by whether or not you agree with me. I'm trying to keep it general.) Perhaps I should add that the book has only gotten 3 reviews -- I'm far from a best-selling author -- and Amazon deleted a very positive review for some reason, so now I've got one positive review and one negative review.
Any thoughts on the above responses, or do you have other ideas? Does the fact that the book is on a controversial subject lead you to give a different answer than if it was, say, a light adventure novel?
- Addendum *
I posted this question over a year ago but someone just posted a new answer, so let me take this opportunity to offer a clarifying comment. (Not saying the new answer indicates a misunderstanding, just that it brought my attention back to this question.)
Bear in mind that I'm talking here about non-fiction books. This is not the same as what I would say about fiction.
If someone posts a review saying that the writing style is poor, like "the logic isn't clear" or "the explanations were difficult to understand", I don't think there's much point to a rebuttal. You can say, "My text is perfectly clear and if you don't understand that's because you're stupid", but that's unlikely to be productive.
But suppose a reviewer says, "Dr Jones 1953 book, 'About Stuff', completely refuted the writer's conclusions." The author might well want to reply that he discussed Dr Jones arguments extensively in chapter 6 and that Dr Smith presented compelling evidence in 1982 that Dr Jones was wrong, and that the reviewer is simply ignoring this. That's the kind of claim made in a negative review that I'm talking about: Not, the writing style is poor, which is largely a matter of opinion and not objectively debatable. But, the writer said X, when in fact I didn't say X. Or, the writer said X which is not true, when in fact I can make a good argument that X is true in a couple of sentences.
I've gotten other negative reviews since and I haven't replied to any of them because, frankly, I don't want to sound like I'm whining about a bad review.