A general strategy you can try here is "respond to amateurism with professionalism". Even if it looks like he's being unreasonable (as it currently seems to you that he is), just suspend disbelief, and, in a completely non-confrontational manner, engage his questions as if they were serious:
I always appreciate constructive criticism [true statement! you don't
have to say that his was constructive, let him think what he
wants...maybe he actually did think it was]. I didn't quite
understand what led you to wonder whether I did enough testing of X.
Did you see something in the results that made you suspect this
specifically? Just want to make sure I'm not missing anything.
Thanks for your input!
Treat it exactly as if it were a sincere attempt to help, that you had a question about. It's possible (though I'm thinking you think it unlikely) that he really is sincere and has something to offer you. If so, wonderful! You get useful feedback from someone smart!
If he doesn't actually have a real question, this technique may make that obvious in a relatively non-confrontational way. Part of the value of it is that you are not simply being the victim of stalker-y behavior, but professionally and politely holding your ground. It is to be hoped that this will lessen the feeling of exasperation that would otherwise normally accompany this kind of childish bullying, if that is indeed what it turns out to be.
There are very intelligent and successful people in academia who are nevertheless woefully underdeveloped emotionally; one way this will be manifest is in the need to tear down other people who seem to be having success, or demonstrating skill, or garnering attention comparable to their own. This can catch one off guard when it's coming from a successful and respected person, because you would think that their success would be all they need to feel good about themselves. That, alas, does not turn out to be true. If it's a chronic emotional problem, you are unlikely to get him to change his behavior; hence the advice to concentrate specifically on blunting his ability to make his problem your problem.
If the cerebral approach doesn't work or becomes too time consuming, you can always set up a filter to send email from that address straight to trash....
Escalation (adding this section after your clarified and expanded examples indicate that mere polite discussion may not work here)
[Fuss level: zero] Gather evidence. You don't have to do anything with it yet. Later steps I will describe will talk about possible uses. It's just good to have in case you ever get into a he said/she said situation and need to be able to back yourself up.
Note that there is already a "fuss", to some extent. Not your doing. You can't choose to have this situation be "fussless", because the behavior is happening in public. The question is to what extent you wish to participate in the fuss.
[Fuss level: low] A body of evidence might be useful for you to have handy if you wanted to petition a site administrator to block him from contributing comments. The sites hosting your research results and facilitating discussion don't want their work compromised by trolls.
[Fuss level: medium] Does he work at an institution with a published policy regarding academic ethics? Spuriously calling into question the validity of your results seems like it would violate ethical standards. It might be enough for you to obtain a copy of the institution's standards, highlight the part that you consider him to be violating, and send that information to him, asking for him to comment on whether he agrees his behavior violates those standards.
[Fuss level: high] Same as 4, but with more fuss. Contact the institution. Ask to speak to someone about the fact that you feel that one of their employees is violating academic ethics. Do they have a published policy? Can they send you a copy? Can you send them some redacted examples and have them confirm that they would consider that in violation? You would like to resolve the matter with the individual privately if possible, but you want to be sure that you are interpreting their standards correctly. Then, email the stalker and tell them that you've been in contact with the ethics office, not having mentioned any names of course, we can resolve this between ourselves, can we not?, and they agreed that the behavior is in violation of standards. Would he be willing to simply stop commenting on your work altogether in the future? It seems it would be better for both of you if he did. You don't have to threaten exposure--you should be careful not to threaten, actually (to avoid any possibility that you could be charged with blackmail or whatever)--just state facts.
[Fuss level: nuclear] Put all the evidence on the web, unredacted, and send a copy to his boss, his wife, and his students. Let them know you really hate to bother them, but you need to ask them if they can contact his psychiatrist because he apparently needs his meds adjusted. (<-- not a serious suggestion, but it was cathartic to type!)
Reflecting on others' answers and comments, I feel that one more section should be added here, because there are practical things you can do to help the situation, from your side only, with zero direct interaction. Although I would still gather evidence in any case (since it might be of value later should a dispute come to the surface), it is possible for you to improve your experience here by improving yourself. It's easy and natural to focus on the fact that he is the aggressor and by rights he should be changing his behavior, and to forget that, completely independent of whether he stops or not you can choose to become better at not being affected by baseless criticism.
Give yourself a mental picture of a small child attempting to attack an adult who calmly holds himself out of harm's way by virtue of having arms twice as long as his attacker. Your stalker has mastered the art of playing the flailing child--you can grow long arms.
Mentally rearranging your perspective on the whole situation can help. You can say "I am going to end up better for this by learning the valuable skill of ignoring trolls. Think what I would have had to pay a trainer for this, and he's providing me all this learning experience for free." It is, after all, entirely possible that you are going to run into other situations in life where the ability to be calm in the face of baseless or childish criticism helps you out immensely, and this is an opportunity to learn that skill in a relatively risk-free environment (as the stalker really has no actual power over you in this situation).
In short, you have, effectively, been focusing on the question "How can I get him to stop this behavior, without making a fuss?" But you can get an excellent result by instead asking the question "How can I get this behavior to stop bothering me?". The second question can be pretty much entirely in your own hands, and the things you do to work on it will help you in other aspects of your life, even enabling you to help others deal with similar issues, etc.
It's more than just ignoring (which, I think it should be said, is fantastic advice if you can do it--I haven't gotten to that point yet personally)--it's a conscious decision to rise above the behavior of the other party. Rather than "he's attacking me and I feel powerless to stop it without escalations that I am not comfortable getting into" it can be "he's attacking me, but I am using this as an opportunity to practice, and therefore develop the power, to not be affected by this behavior". From the outside it may look the same, but mentally, this kind of approach can make all the difference.
I think this is both the best option and the hardest to execute on. In the times when I have implemented such measures myself, I've found it to be incredibly empowering, and even in some cases ended up with unexpectedly good relationships with people that at one point had occasion (they felt, at the time, at least) to attack me, simply because I consciously decided that I didn't need to fight back. I don't think I've faced exactly the same situation as you are here, but I believe this might help you and sincerely hope it does--best of luck.