I'll have to guess at what exactly you mean by "Niven Ring" and "stable"... but here's some "thinking out loud" to get the ball rolling:
I assume you mean a small Ringworld-like structure around Uranus, with some or all of it dipping into the atmosphere. Obviously, it will experience drag forces until/unless the parts in the atmosphere are moving at the same velocity as the atmosphere. But I assume that the structure needs to be moving relative to the atmosphere in order to perform its extraction function. So the structure will be rotating at one speed while the extraction operation is going on, but a different speed if the extraction ever stops. I can't guess which speed would be higher.
The classic Ringworld maintains its shape by virtue of the tension created in the base material ("scrith") by the fact that it is rotating at faster than orbital speed. Even then, it requires active control in order to keep it in position relative to its star. And at less than orbital speed, it would have to have enough compressive strength and rigidity to maintain its shape in order to be considered in any sense "stable".
So the question is, is either (or both) of the two speeds mentioned in the first paragraph large enough to keep the main part of the structure under tension — moving faster than its own orbital speed? I don't know enough about the atmosphere of Uranus or the orbital mechanics around it to calculate the answer.
Also, remember the active control required to keep the structure centered. If this ever fails (at either speed) and the ring goes off-center, the unbalanced drag forces will surely tear it apart. Do you consider this "stable"?