Asking followup questions

Is it OK to ask follow-up questions? So if I asked someone “Would you like some of my ice cream?“, and they said no but I felt there was more to it, and I said something like “should I avoid offering you food?”, would that be OK?
This is often ok, yes.
I think phrasing it as “Would you rather I didn’t offer you food?” would be slighting better. “Should I…” could be perceived as defensive, It could also be read as you wanting them to offer a reason that you could  potentially argue with. “Would you rather…” is clearer about wanting to know what their preferences are.
If you can read body language, pay attention to it when you’re asking questions. If they seem uncomfortable, it’s probably better not to ask additional follow up questions.
(If they’re a friend, it might be appropriate to ask if they’re ok. If they’re a professional associate, err on the side of backing off.)
Sometimes it’s better to back off rather than try to fix things, especially if the problem is that someone is feeling really self-concious.