Only if no one has 270 or more votes.
You and I are talking about separate things. You are talking about the House deciding the race if no one has a majority. I'm talking about what happens on January 3rd (I think) when the House gets together to count the Electoral College votes. They are specifically tasked with the responsibility to determine the eligibility of the candidate that wins. If he is ineligible then the person with the next most votes is declared the winner by the House.
I sent an Email to Tom Price trying to get him to object to eligibility but apparently he didn't.