If your opponent truly believes he did not make the move, he should contact the site support staff. You have no choice but to continue play.
Sportsmanship
It'll settle whether the move was played (though as his Q has disappeared, I guess it must have), but I don't know whether it would prove that it was the move he submitted. And if it did, I don't know if he would accept that!
But my point, really, is am I responsible for putting this right? Can he reasonably accuse me of 'cheating' or 'taking an unfair advantage'?

If you notice, it is not possible for you to make his move, so any move recorded was either done by your opponent or made at random by the computer.
I suspect that if this opponent really believes he did not make the move, he probably either made the move on the board, then opened the Analysis board and found the one he thought he made but didn't change the main board to reflect that change and then clicked Submit without checking to see that his later Analysis move was shown, OR he clicked the Submit button with the wrong move on the main board instead of the Analyze button and didn't realize it. I would guess that the chess.com staff would back you up.

I’m with the rest on this… weather he accidently moved or purposely and trying to reverse I would just forward the complaints to the chess.com staff. There has to be a way to confirm.

Sadly some seem to be slipping into a Machiavellian attitude with regard to sportsmanship. Niccolo Machiavelli is quoted as saying:
"in the actions of all men...when there is no impartial arbiter, one must consider the final result."
Consequently, since "the end justifies the means" for some of our players, it is necessary to throw many of them out for "cheating."
I show my age when I recall being taught under the wisdom of the sports writer Grantland Rice:
For when the One Great Scorer comes
To write against your name,
He marks-not that you won or lost-
But how you played the game.”
I'm currently playing a correspondence game in which my opponent made a one-move blunder: I was able to take his Q with my own, for free. He then posted to say that he hadn't put his Q on that square (I don't know if that is actually possible - I suspect he did so accidentally, and didn't realise it), and was there any way to fix it. I said I couldn't see any, and continued playing.
A couple of moves later, I checked the chat on that board to find that he had (a) suggested I give back my Q, as otherwise the game was spoiled and (b) accused me of cheating for carrying on playing. Not surprisingly, when I first read this accusatory post I didn't choose to do him any favours! But is it bad sportsmanship to play on when your opponent blunders a queen - or other major piece - in a rated team match (*not* a training game)?
I personally think it's the other way around - if you blow a balanced game with a blunder, you should accept it, resign, and offer a re-match if you want one, rather than requiring your opponent to fix your game for you. Asking your opponent to fix it - and then abusing them for not doing so - seems out of order to me. Do you agree?