CBS News White House Correspondent Bill Plante reports that Cheney made it clear with unusual candor that he differs with Mr. Bush on the amendment.
At a campaign rally in Davenport, Iowa, on Tuesday, Cheney spoke supportively about gay relationships, saying "freedom means freedom for everyone," when asked about his stand on gay marriage.
"Lynne and I have a gay daughter, so it's an issue our family is very familiar with," Cheney told an audience that included his daughter. "With the respect to the question of relationships, my general view is freedom means freedom for everyone ... People ought to be free to enter into any kind of relationship they want to.
"The question that comes up with the issue of marriage is what kind of official sanction or approval is going to be granted by government? Historically, that's been a relationship that has been handled by the states. The states have made that fundamental decision of what constitutes a marriage," he said.
Mr. Bush backs a constitutional amendment prohibiting gay marriage, a move Cheney says was prompted by various judicial rulings, including the action in Massachusetts that made gay marriage legal.
"I think his perception was that the courts, in effect, were beginning to change, without allowing the people to be involved," Cheney said. "The courts were making the judgment for the entire country."
Addressing Mr. Bush's position on the amendment, Cheney said, "at this point, save my own preference, as I have stated, but the president makes policy for the administration. He's made it clear that he does, in fact, support a constitutional amendment on this issue."
The amendment to ban gay marriage doesn't have the support to pass Congress, but it is an important issue to conservative Christians whose votes the president needs.
Cheney's comments drew criticism from the conservative Family Research Council, with President Tony Perkins saying, "I find it hard to believe the vice president would stray from the administration's position on defense policy or tax policy. For many pro-family voters, protecting traditional marriage ranks ahead of the economy and job creation as a campaign issue."
Perkins added that if Cheney sees a problem with activist judges, "then how can he not endorse the same solution the president and his pro-family allies have proposed? We urge Vice President Cheney to support President Bush and a constitutional amendment on marriage."
Steven Fisher, spokesman for the Human Rights Campaign, a gay and lesbian advocacy group, said Cheney's remarks show a stark difference with Mr. Bush's efforts "to put discrimination in the Constitution."
"President Bush is feeling the heat. The administration has been using gay Americans to drive a wedge into the electorate. There are millions of American families who have gay family members and friends, who are offended by the president's use of discrimination," Fisher said.
Last month, Lynne Cheney said states should have the final say over the legal status of personal relationships, a comment that came just days before the Senate failed to back the ban.
The Cheneys have two daughters, both of whom are working on the campaign. Mary Cheney is director of vice presidential operations for the Bush-Cheney re-election campaign. She held a public role as her father's assistant in the 2000 campaign and helped the GOP recruit gay voters during the 2002 midterm elections.
During the 2000 campaign, vice presidential candidate Dick Cheney took the position that states should decide legal issues about personal relationships and that people should be free to enter relationships of their choosing.
Sens. John Kerry of Massachusetts and John Edwards of North Carolina, oppose the amendment. The Democratic candidates also oppose gay marriage, but defend a gay couple's rights to the same legal protections as those conferred in marriage.