How should I respond to a reviewer's complaint about self-citation?
The answers here of Solar Mike and corey979 are correct, but let me point out two additional issues.
First, it might be that without your citations you could validly be accused of self plagiarism. Readers of the current work need to be able to trace back the ideas to earlier work. This is why we cite ourselves rather than just reuse old work.
Second, if you want a mental check on whether a self-citation is proper or not, just ask whether you would still be giving this citation if the other paper were written by someone unknown to you. If the answer is yes, then it is certainly proper - even essential - to cite it. If the answer is no, then you should probably forgo.
As for a response, you could simply ignore it. However, if you think that it is affecting the editor's decision, you could point out the self-plagiarism aspect.
6 out of 44 is less than 14%... If the cited works are relevant, such as building on previous results or analysis then there should be no problem.
If you are citing works that are by you but not relevant then that is an issue (I don't think you are doing this but just for both sides).
If the only works you are citing were your own then that may be an issue, but could still be relevant ie further work etc as above.
Perhaps you respond by pointing out the relevance of each cited work and, also consider if the links between the works cited and the current paper need to be strengthened.
The phrase "cheap act of self promotion" might be viewed as offending, and is surely unprofessional (it's just the reviewer's opinion). I'm a hothead, so I would point this out to the editor and ask him to discipline the reviewer.
Overall, self-citations are a way of self promotion – yes, you point the reader to your earlier relevant works in the topic, but you also advertise your previous papers in hope that those that missed them will cite them in their own future articles. Citations are a valuable asset in academia, so it's not surprising authors go after them. Self-citations, however, don't stand on equal grounds as citations – in my field, many evaluations require "number of citations excluding self-citations", so self-citations are not just an easy way to boost ones metrics.
Referring to one's earlier works in the topic is definitely a good thing, showing the author's experience, linking to previous state-of-the-art, and simply telling a story that's behind research ("previously, I've made the analysis in 1D, and herein, for the first time, a 2D analysis is performed"). If your self-citations fulfill any of these roles, I see no reason to remove them.
In the response to the reviewer point out the relevancy of the citations used, like Solar Mike suggests. And the 6/44 ratio is all fine to me – after all, you're the expert in the field, so it's natural you have achievements. If it was 38/44, that would look silly.
Heck, I've seen reviewers flooding their (anonymous) reviews with a list of "suggested" references, orbiting around one author – it's hard not to be convinced about the identity of the reviewer, and think of it as a "cheap act of self promotion"...