Snyder acknowledged same-sex couples "had a legal marriage." But because of the court's stay, he added, the gay marriage ban has been restored.
The governor's move closes the door, at least for now, to certain benefits reserved solely for married couples. The American Civil Liberties Union said more than 1,000 Michigan laws are tied to marriage.
"We did our own homework and I believe this is a reasonable legal position to take based on the available literature and law," Snyder told reporters.
Other elected officials have urged the Obama administration to recognize the marriages for federal benefits. The U.S. Justice Department, which previously said it was monitoring the situation, did not immediately comment after Snyder's announcement.
Dana Nessel, an attorney for two Detroit-area nurses who successfully challenged the gay marriage ban, said Snyder's position is "really an outrage."
"I think each one of those couples should be furious right now, and I'm very hopeful that those couples will petition the court on their own behalf," Nessel said.
Snyder, a Republican who keeps mostly silent on social issues, had said very little since Friedman made his historic decision last week. Snyder said in a 2010 debate that he supported marriage as "between a man and a woman."
Another Republican, Attorney General Bill Schuette, has aggressively defended the gay marriage ban, which was approved by 59 percent of voters in 2004. He said it's his job to oppose challenges to the state constitution.
Snyder said he hasn't had a role in Schuette's legal strategy.
"The attorney general is a separate constitutional officer in our state, and he has the prerogative to make his decisions on that particular issue on his own," the governor said.
Lisa Ulrey and Donna DeMarco were married Saturday at the Oakland County clerk's office. Marriages were also performed in Ingham, Muskegon and Washtenaw counties.
"I'm shocked but not surprised," Ulrey said of Snyder's decision.
"Everyone was on such an emotional high on Saturday," she said. "We felt we were on top of the world. I guess we were naive in thinking the government would be on our side."
Ingham County Clerk Barb Byrum, who officiated at the first Michigan same-sex marriage, said Schuette needs to drop the appeal.