Does my opponent need to prove his creature has morph?
Rule 707.9 says this:


  If a face-down permanent moves from the battlefield to any other zone, its owner must reveal it to all players as they move it. If a face-down spell moves from the stack to any zone other than the battlefield, its owner must reveal it to all players as they move it. If a player leaves the game, all face-down permanents and spells owned by that player must be revealed to all players. At the end of each game, all face-down permanents and spells must be revealed to all players.


There is no way for a game to end without one of these things happening to each face down permanent, so every face down permanent must be revealed one way or another by the end of the game.