Nobody could stop him from doing so. He could do all the right things that are prevented by the boundaries that are self imposed by our society, politics and economy. And since he doesn't represent any nations, his actions would not cause wars between countries or factions.
You are right, logically. But humans don't tend to react logically, but emotionally. They would see that Superman lives in the U.S. and claims U.S. citizenship. Emotionally, people around the globe would blame the U.S. for his actions, and take action against the U.S.
And because Superman can't be everywhere at once, he cannot prevent all of these actions against the U.S., and therefore cannot take irresponsible, no-borders action.