Only if the 12th Amendment applies. But the 12th Amendment doesn't apply - one candidate got a majority of the Electoral votes. If Obama is found to be ineligible then it's the 20th Amendment that governs how his replacement would be determined. And the 20th doesn't contain that restriction.
There is still a chance that the 12th could apply. If Obama’s votes are disqualified that would leave McCain with an insufficient number of EV which would then throw it to the House. You could be right that the 20th would be operative but it is not clear which would be involved not knowing what the Congress would do at the critical moment.