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: 
In the specific case you describe I second what J.M. said. In my opinion the most apt course of action is to state that you do not wish to elaborate. 

This has several advantages: 

1. OP knows there is no point in waiting for you to reply.
2. Others will know that you decided actively not to elaborate further and this might very well inform what they do. It is common for other user to step in and provide additional details. I do this frequently. I do this to avoid the wait until the original answerer comes back. I usually would not if I knew they actively do not want to give more information out of some pedagogical consideration.

Thus, leaving considerations of etiquette it is efficient to be explicit regarding your intentions. 

It depends on context but, generally, I do feel it is a bit impolite not to acknowledge a request. Not to do what is requested is fair game, but at least you should say so.

Of course, such as there can be exceptions, if there were earlier interactions or the answer is explicit about it not being developed etc. 

I also agree with hardmath that if you find yourself often in that situation you may want to look into the choice of the questions you answer. The point of the site actually is for the most part to give ready-to-use answers. This is a vague notion, but it still means that it is in principle not at all unreasonable that somebody asks for further details. 

Certainly, some user behave in unreasonable ways, but one request for further details is pretty reasonable and should be treated as such and not as an attempt to "usurp [ones] time" etc.
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".