As far as I know the UN already has no legal authority over the USA, so this wouldnt make sense... we currently cant and dont veto UN rulings since they have no legal bearing on us unless we want it to (and pass our own separate law to enforce it).
As far as I know no country has any internal system that obligates it to the ruling of international court.
That said I would be very much against giving the UN, or any international court, authority locally. Each country can and should choose to adopt their rullings on a case by case basis.
I wouldnt quite go that dramatic. I mean.. countries could, int heory, accept majority rule. I would be scared of genocide, but I dont think its "required" for such a thing to exist.. the bigger concern is that it enables it at all.
For me I generally prefer local governments having power over larger governments.. at least within a framework. So any sort of global government heads in the opposite direction of what I'd like.