King of Tokyo: Dice resolution -> Yield -> Card buying?
This is what the rules have to say:


  If you are attacked while in Tokyo
  and you don’t want to be there
  anymore (and who can blame you
  – it can be rough!) you can yield
  to the Monster that attacked (he
  has to take your place). You still
  take the damage, however. 


And close by:


  Warning: A Discard card that
  deals damage to you is not
  an attack


Does he think he can leave at any point in the turn after he is attacked?

The rule says "if you are attacked", not "if you were attacked". Everyone with whom I've played has understood the first paragraph I quoted to mean one can only choose to cede Tokyo as one is attacked. There is no chance to leave Tokyo at any point until the monster in Tokyo is attacked again.

Does he think taking damage from a card is an attack?

The second paragraph I quoted makes it clear it isn't.