Absolutely. And his foreign national father already renders him ineligible for the office.
I agree, but the bulk of the country has such a bad understanding of the correct meaning of the term "Natural born citizen" that they grant this aspect of his non-citizenship no weight.
If it can be demonstrated that he cannot even prove he was born within U.S. territory, then he cannot be a "natural born citizen" even by the faulty standards of the majority of the population.