In an official blitz game, what happens if a player hits a chess clock without making a move? I believe that making an illegal move in a blitz game is an automatic loss. But does this count as an illegal move because my opponent didn't really make a move?
You can only make an illegal move by making a move. Pressing the clock without moving is not making a move. So the arbiter can't use that rule to make a decision.
There is FIDE rule 6.13 (chapter 6 is about the clock): "If an irregularity occurs and/or the pieces have to be restored to a previous position, the arbiter shall use his best judgement to determine the times to be shown on the clocks. He shall also, if necessary, adjust the clock’s move counter."
So the arbiter can use his best judgement to reinstate the correct clock times. This is of course unfortunate (it's hard to stay concentrated for the players, especially the innocent player), but the arbiter can't just invent rules.
If this happens regularly (the player is doing this on purpose), there is always article 12.8: "Persistent refusal by a player to comply with the Laws of Chess shall be penalised by loss of the game. The arbiter shall decide the score of the opponent."
An illegal move is completed once the opponent’s clock has been started. The opponent is entitled to claim a win before he has made his own move. However, if the opponent cannot checkmate the player’s king by any possible series of legal moves, then the claimant is entitled to claim a draw before he has made his own move. Once the opponent has made his own move, an illegal move cannot be corrected unless mutually agreed without intervention of an arbiter.
There have been recent (1/1/2018) changes to FIDE's rules that relate to this question.
Previously, in blitz/rapid a player making an illegal move would instantly lose the game.
However, January 1, 2018 revisions to FIDE Laws of Chess acted "to equalize the rules for classical chess on the one hand, and rapid and blitz on the other, with regard to the penalty for making an illegal move."
Before, in both rapid or blitz an "illegal move" (which includes moving with two hands, or not making a move and pressing the clock) would instantly lose the game in case the arbiter intervened or the opponent stopped the clock. Now, in the first instance it will lead to a time penalty and only if a player does it again, he or she loses—just like in classical chess.
Source for quotes on 1/1/2018 change: Peter Doggers, "New FIDE Laws Of Chess For Blitz, Rapid Still Not Perfect," Chess.com, January 3, 2018.
You are correct in saying that in blitz, an illegal move loses the game. In this case, an illegal move is made because your opponent did not move. Hence, if you correctly stop the clock and make a claim to the tournament director, you win the game.
I feel that this would have to be intentional though. I'm not sure what would happen if your opponent accidentally brushed the clock or something. I suppose the decision would be left up to the arbiter.