Bush's call for a constitutional ban on gay marriages failed last year in Congress, but his position was seen as a key factor motivating Christian conservatives concerned about "moral values" to turn out in large numbers and help supply Bush with a winning margin in last week's election.
"If we want to have a hopeful and decent society, we ought to aim for the ideal, and the ideal is that marriage ought to be, and should be, a union of a man and a woman," Bush political aide Karl Rove told "Fox News Sunday."
Rove said Bush would "absolutely" push the Republican-controlled Congress for a constitutional amendment, which he said was needed to avert the aims of "activist judges" who would permit gay marriages.
Renewing his push for an amendment -- despite its slim chances of success -- would be a way for Bush to reward his conservative base. The amendment would face a steep hurdle winning the needed approval of three-fourths of the states.
Other items on Bush's second-term agenda included nominating -- without a "litmus test" on abortion -- judges who would "strictly interpret" the Constitution, and tax reform. Rove said Bush wanted to review the tax code "in its entirety," which suggested a broad-based reform was possible.
Republicans' ability to deliver on their campaign agenda will help determine whether the party can realize its potential to retain a governing majority for decades, he said.
The gay-marriage issue leaped into the campaign spotlight this year after Massachusetts legalized the practice in response to a state Supreme Court ruling, and San Francisco began performing gay marriages in defiance of a state ban.
Ballot measures in 11 states to ban gay marriages all passed last week. Gay-rights groups have vowed to keep fighting for legal protections of same-sex relationships despite the election setbacks.
Bush said last month that he disagreed with a Republican Party platform provision that would also ban civil unions of same-sex couples, and he said states should be able to allow such legal arrangements if they wish.
Rove elaborated on this on Sunday.
"He (Bush) believes that there are ways that states can deal with some of the issues that have been raised, for example, visitation rights in hospitals, or the right to inherit, or benefit rights, property rights, but these can all be dealt with at the state level, without overturning the definition of marriage as between a man and a woman."
U.S. Sen. Susan Collins, a Maine Republican, said a constitutional amendment was unnecessary. "The states are perfectly able to handle this important issue on their own," Collins said on CBS's "Face the Nation."
Asked whether Bush intended to appoint anti-abortion judges to Supreme Court vacancies considered likely to come open in Bush's second term, Rove said the president would not use a litmus test. He said Bush wanted his judicial nominees to be "impartial umpires" who would strictly interpret the law and Constitution.
He played down a conservative firestorm over a suggestion last week by Sen. Arlen Specter, a Pennsylvania Republican, that Bush would have a hard time winning confirmation of any Supreme Court nominees who would overturn the landmark Roe v. Wade decision legalizing abortion.
Specter is expected to become chairman of the Senate Judiciary Committee with authority over judicial nominations.
Rove said Specter has assured Bush that his nominees would receive a prompt hearing and those picked for an appellate court would receive a vote by the full Senate.
Specter said on CBS that he had only been trying to point out that Republicans, while they expanded their Senate control in Tuesday's election, still lacked the Senate votes to overcome a united Democratic front.