In Uno, can you quickly finish a two player game by skipping your opponent's final turn and without saying Uno?
Player 1 hasn't won yet at all. They broke the rules.

Uno says when it's your turn, you play a card. Then it's the next player's turn. That player might have to do something with their turn (like draw cards) but if they have a draw 2 in their hand, they have an opportunity to respond instead.


  Player 1 says [...] they are not obligated to give you time to A) draw 2 cards or B) call them out for not saying Uno.


Actually, that's exactly what player 2 has time to do now. Player 1 played their card. Now it's player 2's turn, and they can choose whether to respond with a draw 2 in hand or not. Players are obviously obliged to give their opponents time to take their turns, and player 1 is given no allowance to just unilaterally decide to skip player 2's turn.

Let's address that bit I skipped over with the “[...]” though:


  Player 1 says it doesn’t matter because they are out of cards and the game is over


Imagine this: at the start of the game immediately after we're given our hand of cards, I just put all mine down face-up in the discard pile and say I've won. You object and say I can't do that. I say it doesn't matter because I'm already out of cards and the game's over and I won.

That doesn't work, does it? I broke the rules, and I definitely haven't won at all. Winning has to occur within the bounds of the game—if I broke the rules, I'm essentially not playing the game, and at that point me winning or losing is neither here nor there.



At the point you reached, you've got some options:


Back up to player 2 having their turn. Player 1 probably draws, on account of being rightly called out for not saying Uno.
End the game (nobody won), shuffle up and play another properly.