NEW YORK — President Barack Obama’s emphatic gay-rights advocacy in his inaugural address thrilled many activists. Yet almost immediately came the questions and exhortations as to what steps should be taken next.

“I was very moved,” said Jon Davidson, legal director of the gay-rights group Lambda Legal. “But there’s a lot more to do in the four years to come. ... It’s not like everything is fine.”

Items on the activists’ wish list include appointment of America’s first openly gay Cabinet member, steps to curtail unequal treatment of same-sex couples in the military and an executive order barring federal contractors from workplace discrimination on the basis of sexual orientation.

The paramount priority for many, however, is same-sex marriage. Never before Monday had an inaugural address conveyed support for marriage equality, and activists now hope the Obama administration will take concrete steps to follow up, including escalated engagement in pending Supreme Court cases.

“Why wouldn’t they decide to stand on the right side of history?” Davidson asked.

Obama broached the broader issues in his speech by classifying the Stonewall gay-rights riots of 1969 as a civil rights milestone on par with those in the struggles on behalf of blacks and women.

Then, alluding to marriage, he said, “Our journey is not complete until our gay brothers and sisters are treated like anyone else under the law, for if we are truly created equal, then surely the love we commit to one another must be equal as well.”

Richard Socarides, a former Clinton White House adviser on gay rights, termed the address “perhaps the most important gay-rights speech in American history.”

Among Obama’s in-person audience were the Supreme Court justices who will be hearing oral arguments in March on two same-sex marriage cases. They will be considering both California’s constitutional ban on gay marriage and provisions of the 1996 Defense of Marriage Act denying federal recognition to same-sex marriages, which are now legal in nine states and the District of Columbia.

The Obama administration already has said those DOMA provisions are unconstitutional and is no longer defending them, leaving that task to a legal team hired by Republicans in the House.

Gay-rights activists say the Justice Department could take further steps in that case, notably by filing papers with the high court aimed at placing an even higher burden on DOMA’s defenders to justify the government’s unequal treatment of same-sex couples.

Activists also hope the administration will file a friend-of-the-court brief in the California case, joining with those who argue that the 2008 Proposition 8 ballot measure banning gay marriage in the state violated constitutional guarantees of equal protection.

Fred Sainz of the Human Rights Campaign, a national gay-rights group, said filing such a brief would be a “natural extension of the inaugural remarks,” which he depicted as “an incredibly eloquent equal-protection argument.”

Political repercussions will probably be discussed further before the White House makes a final decision on the Supreme Court cases, Sainz said. But he suggested the administration had realized — after Obama’s re-election — that advocating for same-sex marriage is “both morally right and politically right.”

Obama has long portrayed himself as a gay-rights supporter and played a key role in ending the “don’t ask, don’t tell” policy in 2011 so gays could serve openly in the military. But only last year, after what he described as a process of “evolving,” did Obama come out publicly in favor of gay marriage, and even now legally married gay couples in the military are denied some important benefits accorded to heterosexual married couples.   

On Tuesday, the White House directed to the Justice Department questions about whether the administration would file a friend-of-the-court brief in the Proposition 8 case. The department declined to comment.

Meanwhile, the defenders of Prop 8 filed their opening brief with the Supreme Court on Tuesday, arguing that the justices should allow public and political debate over same-sex marriage to continue rather than impose a judicial solution.

Other opponents of same-sex marriage took note of Obama’s inaugural remarks, in some cases with alarm and anger.

“Their implications are morally devastating for the definition of marriage,” wrote Denny Burk, a professor of biblical studies at the undergraduate arm of the Southern Baptist Theological Seminary in Louisville, Ky. Burk, in a blog posting, contended that Obama’s rationale for legalizing marriage for gays could be extended to polygamists as well.

Brian Brown of the National Organization for Marriage, which has campaigned against same-sex marriage in many states, criticized Obama’s decision to raise the topic in his address.

“A presidential inauguration should be a time for the nation to come together,” Brown said. “Instead President Obama chose to voice his support for a radical agenda advanced by some of his biggest campaign contributors to redefine marriage for everyone.”