Republican Jan Brewer said the bill would have allowed officials to judge who is eligible to run for office.
A lingering "birther" conspiracy theory asserts US President Barack Obama was not born in the US and is thus ineligible to hold the office.
But the bill's Republican backers insisted it was not aimed at Mr Obama.
The Arizona legislature was the first to pass such a law.
It would have allowed the state's top election official, the secretary of state, to determine whether candidates met citizenship requirements to hold the office of president.
"I do not support designating one person as the gatekeeper to the ballot for a candidate, which could lead to arbitrary or politically motivated decisions," Ms Brewer said in a statement.
She was secretary of state until she became governor in 2009.Mr Obama, seen here with his Kenyan father, was born in Hawaii and spent his youth there and in Indonesia
The US constitution requires the president be a "natural born citizen", a clause widely interpreted to mean born in the US or in some cases to US citizens abroad.
Mr Obama has released a certificate of live birth showing he was born in the US state of Hawaii, where officials vouch for its authenticity.
But the "birthers" claim Mr Obama was born in Kenya, his father's place of birth, or perhaps in Indonesia, where he spent several years as a child.
The bill would have required national political parties to submit affidavits affirming their presidential candidates are "natural born" citizens and to provide a "long form birth certificate" listing the name of the hospital and the attending physician.
If the candidate did not possess that document, the candidate could provide a baptismal or circumcision certificate, hospital birth record, mother's post-partum medical record or early census record.
"I never imagined being presented with a bill that could require candidates for president of the greatest and most powerful nation on earth to submit their 'early baptismal circumcision certificates' among other records to the Arizona secretary of state," Ms Brewer said.
The birther conspiracy has simmered at the fringes of American politics since before the 2008 presidential election, despite repeated assertions by Hawaiian officials that Mr Obama's birth certificate plainly states he was born in the US state.
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