Imagine that I write a paper about a controversial topic like global warming denial, the link between vaccines and autism, why different races have different IQ. After publication, the paper gets the attention of mass media, and as a counter measure, serious experts start explaining why the paper is completely wrong.
Would that count as citations towards a higher h-index?
Yes, you can!
But that it is possible is by no means to say that it is ethical, practical, wise, or otherwise commendatory. I would be especially concerned about becoming known as the 'person with a kooky idea' rather than as a serious academic researcher.
The question becomes, "It is possible to write on a very controversial topic, create a media firestorm, attract a lot of attention, increase your h-index, and still keep your credibility intact?" The answer will be highly variable dependent on the validity of your research, your previous reputation, and the sheer capriciousness of luck.
First, assuming that your work is valid, even if you have proven the viability of a very controversial position, your work is likely to attract some negative attention as well as attempts to disprove your research (or you-- ad hominem attacks are unfortunately common). However, if your work can and does stand up to scrutiny, all the brouhaha may actually work in your favor-- you have proven a controversial theory to be true, your h-index will increase and your credibility is not only intact but also bolstered by your success.
On the other hand, if your work does not stand up to scrutiny (which seems to be the scenario you are picturing), you will have made a public fool out of yourself and the slight increase in your h-index will be more than offset by the decrease in your credibility. Neither the counter-moves of serious researchers nor the attacks of fanatics are likely to help your academic career, especially if your work cannot stand up under scrutiny.
So, write a really bad, but controversial, paper only if you are willing to sacrifice your credibility for the slight increase in your h-index.
h-index counts citations regardless of the content of those citations, so citations by people criticizing the paper, disagreeing with it, or pointing out that it's nonsense do still count as citations.
(As a plan to improve one's h-index, this seems like a bad plan for a number of reasons. As a concern about the meaning of h-index, it's a concern, though there's room to argue about whether this sort of situation is common enough to matter.)
Not really: note that by definition of the h-index this paper can increase your h-index at most by one, unless you are lucky enough to get citations of the type described by David Richerby in the comment below ("X, despite making significant contributions to the subjects A [1-3] and B [4,5,7-13], has some unorthodox opinions on the subject C [6]").
However, attracting this kind of citations is very field-dependent, and I doubt that even if this strategy works out, be it with a single paper or in David's way, it would really pay off, especially given the losses in reputation.
No, you probably can't, because for it to gain a lot of attention it needs to be intriguing in some way. Simple rubbish isn't; there's plenty of that already, and the peer review process screens out most of it. You could try some huge publicity campaign, but if you're that good at publicity, maybe you're in the wrong field?
Finding just the right balance of plausibility, tension, incorrectness, and publicity is very hard. One indication that this may be the case is that the number of highly-cited bad/controversial papers is much smaller than the number of highly-cited good papers.
Just write a good paper. It's easier (not easy!) and more useful.
Most bad papers are ignored not cited for being wrong. You would have to get it into a good journal and get people to praise it etc. before other researchers will think it is worth critcizing it.
In my experience, most papers of this nature tend to be written by senior academics near the end of their careers, and so it has little impact on their h-indices as they already have a sufficient number of papers with more citations than the controversial paper is ever likely to attract. Less senior academics at the start of their careers (where it might have an impact on their h-index) tend to be more circumspect and careful (as their lack of experience tends to make them more self-sceptical). As a scientist, self-scepticism is a vitally important quality to be carefully nurtured.
I just want to add that your examples are not likely to succeed. If you attract media attention, the debunking papers will be published in newspapers, not in peer reviewed journals, and it will not count towards bibliometric indexes. For it to work you would need to attract attention in your field, and therefore, fool experts.
This said, there are some exceptions, one "good" and one "bad":
This is a clever idea, never thought of it.
You are famous for the wrong reason. Of course, what's being worse than being talked about is NOT being talked about. Unlike Hollywood, publicity does not equal fame, nor does it generate grants.
