In the US, would it be possible for a re-vote for the President to be called for if, for example, it was highly suspected or confirmed that the first vote's results were rigged?
The Constitution provides no mechanism for a re-vote of the Presidential election. However, it does provide three mechanisms by which a rigged popular vote can be over-ridden.
The Electoral College can collectively ignore the popular vote. Electors are nominally pledged to vote for a given candidate, but if enough of them act as faithless electors, it can change the outcome of the election.
Congress can reject the electoral votes from states with voting issues. Doing so requires a majority vote in both houses of Congress; this happened in the 1872 election, when the votes from Arkansas and Louisiana were rejected due to voting irregularities.
Congress can reject the election entirely by not recognizing any of the electoral votes, sending the election to the House of Representatives and Senate as provided in the Twelfth Amendment. It's unclear what "the persons having the highest numbers not exceeding three" would mean in the context of an election where zero electoral votes were cast, so it's likely that, in practice, sufficient electoral votes would be accepted to generate a tie.
Short answer: depends on what you mean by revote.
This answer gives a way of doing what you want within the framework of the constitution.
Presidential elections are actually decided by the electoral college so the election doesn't actually take place until December. Each state is free to choose their own method of elector appointments. Hypothetically a state legislature could pass a law saying that the first election in November would no longer determine the presidential elections, instead a future election would. If all the states do this, you could have a "revote".
States are free to set up their own form of republican government and thus elections to state positions are much more amenable towards experiments like revotes or something similar (take a look at recall elections).
Last type of elections are Senate and House of Representatives Elections. These types of election results are generally more difficult to create a revote condition. Essentially, the people who elect these people cannot directly cause a revote, but the House and Senate can refuse to seat members through the constitution's clause allowing each chamber to judge their own elections. Thus this way each chamber can force a specific House or Senate seat to be voted on.
Each method is probably extremely politically destructive, (to incumbents' chances for reelection), and will probably not occur.
With our adversarial court system and heavily partisan politicization of the judicial system over the past few decades, I think it's highly unlikely that such a thing would happen. You'd have dueling, possibly politically-fueled contradictory rulings going up through the court system, definitely winding up at the Supreme Court.
In general, SCOTUS tends to hand down more modest rulings that might look like, if there was evidence that the elections, overall, were tainted - "The results of the election are completely suspect and undermine fundamental confidence in the basis of our democracy. However, there is no mechanism for extending the Electoral College deadline, so the voters have to suck it." (see Herrera v. Collins where SCOTUS ruled that proof of innocence is not reason enough to overturn a death penalty sentence, if correct procedure was properly followed throughout the process).
Our overly litigious society and rule of law precludes common sense or common good overriding the actual pedantic reading of the letter of the law, even if an obvious case for common sense should arise.
That might seem to thwart "justice" in some cases, but if that wasn't the case, you'd have an imperial judiciary that could write law and would be above any checks and balances that the US system is built upon.
No one ever envisioned that any system built would be perfect, so they tried to limit the amount of damage that could be done by splitting the powers up and limiting what any branch could do. They clearly felt that the danger from an unchecked branch of government was much greater than having to live with the results of a flaw in the system for a period of time.