It's the electoral college that elects the president. Therefore, it is the states that elect the president. It's just that the state laws tie the electoral votes to the votes of the people in that state. So your right to vote for President doesn't exist in the federal constitution, it's in your state constitution.
But it's important to note that the state has a right to distribute its electoral votes any way that it wishes. It can do it as winner take all as most states do, or it can divide it by congressional district which Maine and Nebraska do. Article II, Section I of the Constitution gives the states this power. The federal government isn't really supposed to be involved in how a state conducts its elections. The Supreme Court has only allowed very few exceptions and only when it is clear the state is trying to violate equal protection.
For example, there has been talk in Michigan and a few other states to divide the electoral vote up by district which would actually benefit Republicans. If the state decides to do that....then there is nothing anyone can do about it. It's perfectly Constitutional.
This is not really that abnormal. The United States is not a direct democracy, we are Constitutional Republic. Also, in most other democracies around the world, the head of state isn't directly elected either. They are elected usually by a coalition in the parliament or legislative branch.
But no, the election can't be cancelled. The Constitution calls for someone to elected every 4 years. Now, there are ways that the person elected didn't actually win. And that's happened before, legally. But in each of those cases the House selected the President. You can't cancel an election because you don't like the results and try again. It doesn't work that way.
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