I obviously am no expert but to me the situation seems as follows:
1. Illegal move from Inarkiev.
2. Carlsen can win by claiming it, but (instinctually or otherwise) moves his king out of danger instead.
3. Inarkiev claims that Carlsen has then made an illegal move, and claims he (Inarkiev) deserves a full point.
In this situation, IMO Carlsen missed the opportunity to claim illegal move (but made a legal move) so the game should either proceed after 28. Kd3 or be brought back to before 27…Ne3+.
I'm not sure what the protocol is exactly, but assuming "position is legal" == "legal move," then game should resume after 28. Kd3, since the game is once again legal.
Inarkiev seems clearly in the wrong, particularly as he's losing and (I'm assuming, given that he later refused to play) intentionally played out this sequence of events to try and secure a win.