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 believe that simply choosing not to reply is a bit rude. I would support leaving a comment such as "Please put more effort into this question. I would rather not post a full solution" instead of not replying.