Gov. Jan Brewer vetoed the bill on Monday saying it went too far.
The bill would have made Arizona the first state to pass such a requirement. Opponents had warned the bill would give another black eye to Arizona after last year's controversy over the state's illegal immigration enforcement law.
The veto comes as President Barack Obama has been hounded by allegations that he was not born in the U.S. Those allegations have been repeatedly discredited.
The U.S. Constitution requires that presidential candidates be "natural-born" U.S. citizens, be at least 35 years old, and be a resident of the United States for at least 14 years. Opponents questioned whether Arizona's bill would have added additional requirements.
The bill's sponsor, Republican Rep. Carl Seel of Phoenix, said he was disappointed by the veto.
The Associated Press contributed to this story.