What happens if other countries contest the US election? Probably nothing.
Somebody mentioned Yugoslavia, but they might as well have said East Timor or just about any country where a dictator or despot or oligarchy, not to mention the corresponding lack of free elections, has been in place for years. The people in such places are generally not used to free elections, secret ballots, the lack of intimidation by the current ruler’s heavies, and other things that we in the Western democracies take for granted. In such places, the UN sends in observers from stable democracies (not just the US, but also places like Canada and Australia) to make sure that the rules are followed and the election is fair. When fraud and intimidation occur, the US and other Western observers complain, and rightly so.
But the US has a long tradition of recurring elections. Most Americans are familiar with the voting process, and know their constitutional rights. Joe and Jane American Voter are not physically threatened on election day by government hit squads, nor are they new to voting–they have done it before (though there are always people voting for the first time), and they know what to expect.
Because of this, the US and other stable democracies don’t need UN or outside help in running their elections. They can do it on their own, and if some kind of questionable activity occurs–as in this US election–the stable democracies can take care of it themselves by looking to their respective constitutions and due processes of law.
Regardless of who wins this US election, the American military won’t stage a coup d’etat, the people won’t take to the streets in open revolt, and when the count settles, someone will have been elected fairly by the rules laid down in the US Constitution. (Assuming the weird Palm Beach situation is resolved, of course.) Other countries’ help in keeping things in the US fair and calm is unnecessary, in other words.
Knowing this, other nations really don’t have a cause for complaint if they don’t like whoever wins this US election. The other countries may have concerns over who wins, because it might affect how the US deals with these places, but they cannot do much of anything if they don’t like the eventual result.