Question: Suppose that I posted, as an answer, a fairly good hint to a question, but the OP still can't solve it on their own, and they ask me for further details. Would it be rude if I did not reply?

The only reason I'd choose to do that would be foreseeing that it would lead to me eventually giving out a detailed, complete answer, and I don't want to do that because it is the kind of questions where the details can be easily filled in if the student understands the underlying material well.

Answer: 
Nope. Nobody has the right to usurp your time. In fact, it is rude of someone to expect that you drop everything to answer every single follow-up question they may possibly have. You are here for your enjoyment, not out of a debt to anyone.
I understand your point that sometimes you feel that the student is not giving a minimum effort, and it is true that you have no moral obligation to reply and elaborate. The best course of action however, in my eyes, as a student, is to reply by saying something along the lines of: 

"what I have written is a direct basic application using the definition of this or that, revise them and I promise the hint should largely be sufficient".