"Nu ai nimic de făcut?"
Translation:Don't you have anything to do?
The supine is formed by Preposition + Participle.
I thought you were asking about the formal grammar specification of the supine and the participle, and I cannot vouch for that. (search "substantivizare supin" ...)
However, I think I now better understand your point of view as a foreign learner. For practical purposes, yes, the participle and the supine are always derived in the same way from the root of the verb, no need to remember anything extra :D.
And yes, your example is correct.