Is there a way to request a game be aborted on Chess.com?

No. Is your opponent on vacation?

I think you can only abort a game within the first two moves or so.

Wish this were extended.

Nothing's worse than pushing out e4 to start the game, seeing e5 come in reply, popping out Nf3, and having black come back at you with something like h6.

Especially at three day (or longer) time controls, you know you're looking at like a month long pointless slog in a game that can't conceivably give you any enjoyment.

That's kind of anti social.
People have to right to play whatever you want. You can't just end a game because you do not like your opponents playing style.

You could always resign. 

Don't think you can Mike after a couple of moves. Had an opponent that started swearing like a sailor after 3 moves & I couldn't do anything but alert the powers that be to the problem. He suddenly disappeared from the rolls so that might be an alternative.

Start from a position that precludes h6.