Each country should elect its leaders. I do not envision a world government ever developing. The United Nations was designed to be a forum for dialogue among the countries of the world, but it has basically been unsuccessful and always will be since it has no binding authority.
So the short answer is no, the world should not vote to elect the president of the United States. They can offer their opinions, though Social Media and other means, but the election of our leaders is a right reserved to the citizens of the United States.