An incorrect country of birth is a fairly major error on a passport. You should definitely get this error corrected before attempting to use your passport.
To correct a US passport, you would submit form DS-5504.
Q. IS THERE A FEE ASSOCIATED WITH THIS FORM?
A. There is no fee associated with the use of this form unless expedited service is requested (see below). Your re-issued passport book and/or passport card and any documentary evidence submitted to Passport Services will be returned to you by priority or first class mail, unless overnight delivery is requested.
Credit:stackoverflow.com‘