If you can find a way to work this out with this person and make this problem go away, that's great. Please make every effort to do that. But I'm going to assume that you can't and that this person keeps making trivial complaints to HR and that this is a serious issue that affects your ability to work. If not, find a polite solution.
You need to make HR do their job. There's a complaint against you. HR is supposed to investigate it and make findings. Instead, it sounds like they're trying to placate the complainant and make the problem go away. They might think they're doing everyone a favor, but they are not doing their job.
Send them a written request for the findings they made with regard to the complaint. Specifically, did they find the complaint substantiated? Did they find that the complaint was made in good faith? Did they find that the complaint was not made in good faith and was an attempt to bully or harass you? (Don't suggest that it was. Just ask them if that's what they found. It's their job to investigate, so they should have either found that or not found that.)
If HR finds that the complaints are not made in good faith, they should deal with the person making the bogus reports. The tough case is if they find that the complaints were made in good faith despite your opinion that they are obviously bogus complaints and where they are really affecting your ability to have a peaceful work environment.
If that happens, notify your manager that you are looking for a new job because you cannot tolerate the bullying and hostile environment that you are getting in the form of these harassing complaints and failure of HR to make appropriate findings and make the harassment stop. Your company will probably get lawyers involved at this point, so keep complete documentation including names of people who can substantiate your claims.
Again, please try to avoid escalating this situation if at all possible. It will be better for everyone, especially you. But if you do need to escalate, that's how.
While it is generally illegal to retaliate against someone for filing a discrimination or bias complaint, in most jurisdictions it is legal to retaliate after a fairly-made finding that the complaint was not made in good faith or was made to harass or annoy. It is not sufficient that the complaint be found to be incorrect or not substantiated. There must be a positive finding that the complaint was not made in good faith, that is, that the person was not really offended or that any such offense was trivial and that the real purpose for the complaint was as a form of bullying or harassment.
Whatever you do, do not pretend to take offense at trivial things. Do not mistreat a fellow professional. Do not retaliate in any way, other than with HR. If you genuinely feel that these bogus complaints and poor HR handling are making your work environment hostile, say so. If not, you ultimately have to live with it or find a new job.
tl;dr: If this doesn't really bother you and you've just worked yourself up into a self-righteous frenzy, just let it go. It doesn't matter if you're right or wrong. You have a co-worker with an annoying quirk you can live with. I'm sure you have annoying quirks too, like making beard jokes. But if this really does rise to the level of you being bullied or suffering a hostile work environment where you live in fear of the next bogus complaint, use the procedures your company has to deal with that. And do not let HR bully you either or placate a bully, call them on it.