I have two players in the hand. On the river player one makes a bet. Player two does not call. In the meantime player one flips up his two whole cards and one falls off the table. Does player one still get awarded the pot since nobody called him on the river?
You're unclear when you say player 2 "doesn't call". Do you mean that he actually made some motion or verbal indication of intent to fold? If so, then the moment that happened, the hand is over and player 1 became the owner of the pot. What happens after that is irrelevant.
If that's not the case, and player 2 still has cards in his hand and has not acted, then the dealer's job here is to simply look at player two and wait for him to act without saying a word. There are three possible outcomes: he folds (and player 1 wins), he calls (and player 2 wins, because player 1 no longer has a legal hand), or he asks the dealer/floor for a ruling before indicating intent. In that third case, it is up to the floorman to do what he thinks is fair. The dealer must be careful not to influence player 2 in any way--he must remain neutral.
In that third case, the floorman will probably tell player 2 something vague like "I'll rule when the action is complete. Please act on your hand". If Player 2 asks specifically whether player 1's hand is dead, he can say "Player 1 cannot win a showdown with an incomplete hand. Please act on your hand." If player 2 then folds, player 1 still wins the pot (no showdown needed). If player 2 calls, he wins, though the floorman might decide to return the last bet and call depending on the situation (for example, if someone else had mentioned that player 1's hand was dead and influenced player 2's call).
In normal conditions the answer should be 'Yes the player one get's his pot awarded.'
But as i always say, some casino/home games have strange rules and you should know them before you sit in a table.
If the player showed his hand and at least 1 other player still act, most of the time the last user can decide his action after seeing both cards. It's like the game keeps playing but the player who saw the cards has a big advantage because he knows where he's at.
At that point the last user could try bluff the other player or just abandon the hand because knows he's behind.