"The DM does it" is only true on one level. While it's possible for the DM to rule on this directly ("I, by the powers vested in me by WotC, declare your paladin Fallen!"), it makes more sense for the DM to role-play the authority who makes this decision. So that does raise the question of who that authority should be. It could be the paladin's god, or the paladin's church, or possibly even the paladin's own conscience. Perhaps paladins get their powers from their faith in their own righteousness, and they fall from doubt...

But I think it's best if the god decides. Role-playing the paladin's god also gives you a nice buffer against the heated arguments that might follow. Instead of a discussion about real-life ethics and how they might apply in D&D, you'd have a discussion about the god's nature and personality. And obviously the DM has final say on those. So instead of "I rule that what you did was evil", you have "I rule that your god no longer approves of you".

(And, I should note, I don't mean "Have the god rubber-stamp the DM's decision". Gods are immensely powerful, but they still have personalities and goals, and so they can be role-played.)