I think the question is "why would" he allow this, not "why won't he".

To prove to the American electorate that he is Constitutionally eligible for the Presidency of the United States.  He is, in effect, applying for a job, and the fact that he has put himself out there as a public figure subjects him to a whole different standard.  He could release a redacted copy to the press to put any rumors in that regard to rest.  I didn't say anything about putting it out on the Internet.