Not in regard to the eligibility of anyone who is elected President by the Electoral College. There is only one body who can rule on whether that person is eligible and only that body can declare him ineligible. I don't think you are going to get enough votes in that body to declare Ted Cruz ineligible.
This is a non-issue, but it somehow has become the central focus of this Forum.
Were SCOTUS to say that, i.e., De Vattel’s Law of Nation’s held sway over the Framers and that is what they understood to be a “natural born citizen,” i.e., someone born within the jurisdiction of two citizen parents (unless serving the nation abroad), anyone not meeting that requirement wouldn’t make it to the electoral college stage, so the House wouldn’t be brought into play. As it is anyone with even a claim to citizenship by today’s understanding can run, be elected and serve as POTUS.