When he is in America, Superman fights for Truth, Justice, and The American Way. But he doesn't stop battling crime at the U.S. border. He's busy righting wrongs all over the Earth and around the Universe. Different jurisdictions have different laws. Does Superman have to abide by those laws or does he feel free to use his own set of rules?
From my understanding, Superman isn't the judge jury and executioner. He's just the guy who stands up for things he thinks are obviously right.
I've seen a few times when Superman will stop a person he thinks has committed a crime, but then, the person doesn't get convicted of anything, so Superman doesn't press the issue and lets whatever government make their own rules.
I also think that Superman is versed in American laws enough that he has the ability to fight obvious crimes (bank robberies, heists, etc) and knowledgable enough to know when his services aren't needed (corporate espionage, insider trading, etc.).