Aides say Brewer is worried about corporate backlash.
Apple, which is currently building a plant in Arizona, is just one company that has voiced its opposition to the bill.
American Airlines and PetSmart are among the others.
"Our associates, our communities and our customers all deserve equal respect," says PetSmart CEO David Lenhardt. "This bill undermines it and goes against that."
Backers of the bill say it protects the religious freedom of business owners who oppose gay marriage.
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