Based on the US Constitution, is a president ethically (if not legally) bound to demonstrate that he meets the qualifications listed in that Constitution on which his oath of office is based?
I don't know that I could prove from the Constitution whether the legal burden of ensuring it lies more heavily on the candidate himself, on the state officials who determine ballot access, on the Electors, or on the Congress when they certify results. But yes, certainly it's reasonable to expect him to provide adequate evidence of eligibility.