Just a thought.
How would you play against this h-pawn thrust?
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The move isnt really bad. My DB has about 140 games with 2 h4 and its scoring 51% and I was surprised that a lot of the games had 2400+ players on the white side.
Interesting. I hadn't thought of 2...h6. That does seem to hinder any plans of white playing h5 anytime soon, though black has inflicted some damage on his own k-side, and whites plan by pushing the h-pawn was to do just that. I'll have to study this variation further.
In the first game (2...h5) black didn't play the way I would have liked :) but there was nothing wrong with what he did. You won because you outplayed him not because of the opening. In general he didn't respect your attack and failed to generate any counter play.
The 2nd game (2...Nf6) was more like what I wanted to see from black (He plays d5 and c5). I liked your play too, it was interesting to me. And let's face it black is fine and is actually pressuring white's king more than the reverse is true. The game ends abruptly... did black run out of time?
The third game (4...c4) I think black is offended by your moves and decides to punish you right away with aggressive play :) I would probably have played this in blitz (4...c5) But this really backfires and you quickly get a nice position.
And wow your play in this game is very nice IMO (maybe your opponent was weaker so it's easier for me to see?) I'm surprised you're only rated 1900 something on chess.com. Either it's a new ID or you're on your way up in any case 
After the moves 1.e4 g6 my intuition tells me the reply 2.h4?! should not be good. It looks wrong. It disobeys opening principles and obviously creates weaknesses on the k-side. Also, after 1.g3 The move 1...h4?! likewise looks poor, and I've not seen anyone bother to mention the move in any analysis, suggesting it's not worth consideration, because no strong player would make such a move. But how should the player of 1.g3 (or 1...g6) best go about refuting this h-pawn thrust? (or is the move sound?)
I have played this h-pawn move in three (serious?) games so far, and have not yet lost.
My games did not actually arise from the above moves. I started the games with 1.a3 and my opponents replied 1...g6 then for fun I went 2.h4. I give the games below: What plan should my opponents have adopted to exploit this wrong looking move?
1.a3 g6 2.h4?! h5
2...Nf6 3.d4 d5 4.Bf4 Bg7 5.e3 c5 6.Nc3 cxd4 7.exd4 Nc6 8.Be2 0-0 9.h5 Ne4 10.Nxe4 dxe4 11.c3 e5 12.dxe5 Nxe5 13.hxg6 hxg6 14.Be3 Bf5 15.Qb3 Qc7 16.0-0-0 Nd3+ 17.Bxd3 exd3 18.Nf3 Rfe8 19.Rh4 Rac8 20.Bd4 g5 21.Nxg5 Rcd8 22.Qb5 Bxd4 23.Qxf5 Bxc3 1/2-1/2
After 2...Bg7 I planed 3.h5 I'm thinking (rightly or wrongly) 'The dread of every player of a fianchetto is to see the rook pawn hurtling towards then, so why not unnerve them immediately, and go straight for the throat'. Rybka4 sees nothing wrong with the position after 3.h5, though some would argue, chess engines are no good in openings.
3.d4 Bg7 4.e4 d6
4...c5 5.dxc5 Nc6 6.Qd2 Nf6 7.Nc3 Bh6 8.f4 e5 9.Nge2 exf4 10.Qd3 0-0 11.Bxf4 Bxf4 12.Nxf4 b6 13.0-0-0 bxc5 14.Qd6 Re8 15.Bc4 Ne5 16.Bd5 Nxd5 17.Ncxd5 Nc4 18.Qxc5 Ba6 19.Qc7 Kg7 20.b3 Nxa3 21.Qc3+ Kh6 22.Nxh5 1-0
5.Nc3 Nf6 6.Bg5 Nbd7 7.Nf3 0-0 8.Qd2 c5 9.0-0-0 cxd4 10.Nxd4 Nc5 11.f3 Bd7 12.Bh6 Rc8 13.g4 Kh7 14.Bxg7 Kxg7 15.g5 Ne8 16.f4 Qa5 17.Qe3 Bg4 18.Bh3 Bxh3 19. Rxh3 Nc7 20.Nd5 Nxd5 21.exd5 Qb6 22.f5 Na4 23.b3 Nc3 24.fxg6 Nxd1 25.Nf5+ Kxg6 26.Nxe7+ Kg7 27.Qxb6 axb6 28.Nxc8 Rxc8 29.Kxd1 Rc5 30.Rd3 f6 31.gxf6+ Kxf6 32.a4 b5 33.axb5 Rxb5 34.c4 Rc5 35.Rf3+ Ke5 36.Kd2 b5 37.Kc3 bxc4 38.bxc4 Ra5 39.Re3+ Kf5 40.Re6 1‑0