No it won't. A passport only certifies citizenship or nationality, not "natural born" citizenship. Arnold the Governator undoubtedly has a US passport, but he's not natural born, nor eligible to be President.
I take that back, the passport shows place of birth, and so is sufficient.
Of course you have to submit a birth certificate to get a passport, or other official documentation of place of birth such as: (From state department website):
Previously issued, undamaged U.S. Passport
Certified birth certificate issued by the city, county or state*
check box Consular Report of Birth Abroad or Certification of Birth Naturalization Certificate
Certificate of Citizenship
*A certified birth certificate has a registrar's raised, embossed, impressed or multicolored seal, registrars signature, and the date the certificate was filed with the registrar's office, which must be within 1 year of your birth. Please note, some short (abstract) versions of birth certificates may not be acceptable for passport purposes.
A quick search did not reveal which short form BC's were and which were not acceptable, except that California's and Texas' are not.