After Desert Storm, I seem to remember there was a minority who wanted to continue until Saddam was out of power, but most people wanted to bring our boys back home, provided Iraq wasn't causing any more trouble. I was kinda young during this time, so if I'm incorrect in my recollection, feel free to correct me.
I don't think we would have gotten UN support to keep going once the primary objective (liberating Kuwait) was complete.

I don't think UN would have complained too loudly, if at all. Methods Coalition forces employed were rather far from impeccable (remember Basra road with its 'collareral damage', fabricated news of Iraqi soldiers plundering matenity ward of Kuwait hospital leaving babies on the floor etc), but there was virtually no outcry and that was basically considered (perhaps correctly) as grim but acceptable price to be paid for liberation of Kuwait (whatever that might mean, given the regime there).

However, US needed Saddam Hussein at the time: someone had to be a balance to Iran, keep Shiite minority in check, help Turkey with its 'Kurd problem'... (Speaking of the later, 'no fly' zones don't apply to Mi-24 helicopter gunships; Kurds felt that.)

So, no, I don't think US would have been stopped if it pressed on a decade ago. It just wasn't in what Bush Sr. considered national interest.