WASHINGTON — The Supreme Court seemed reluctant Tuesday to find a constitutional right to same-sex marriage in a landmark case challenging California’s Proposition 8, the initiative banning such unions that voters approved in 2008.

The justices explored grounds for a narrower ruling that would make same-sex marriage legal in California but not require it in other states that now ban it. A decision is expected by the end of June.

Justice Anthony Kennedy, considered the pivotal vote, said that extending same-sex marriage rights to all 50 states could channel the court into “uncharted waters”

Still, Kennedy worried about the estimated 40,000 California children in families headed by gay or lesbian couples whose unions the state deems separate from, but equal to, marriage.

Kennedy said he was concerned about “what could be a legal injury, and that’s the voice of these children.”

California officials, including Gov. Jerry Brown and Attorney General Kamala Harris, refused to defend Prop. 8. Four of the initiative’s five original sponsors, represented by former Reagan administration Justice Department lawyer Charles Cooper, are defending it.

Several justices explored whether these sponsors had cause or “standing” to be in court, since they were not personally harmed after lower courts struck down the initiative.

Ted Olson, a top Republican lawyer and former Solicitor General in the George W. Bush administration who argued the case for Prop. 8 opponents, asked for a broad ruling that would find a constitutional right to same-sex marriage on equal-protection and due-process grounds.

Justice Sonia Sotomayor asked Olson, “Is there any way to decide this case in a principled manner that is limited to California only?”

Olson said the court could let stand U.S. District Judge Vaughn Walker’s decision finding the initiative unconstitutional, or a similar but narrower ruling in the Ninth Circuit Court of Appeals.

San Francisco City Attorney Dennis Herrera, who joined the suit after defending former Mayor Gavin Newsom’s 2004 issuance of marriage licenses in San Francisco, said the court has a “full range of options,” from making marriage legal just in California, or in some states that now offer only civil unions, or extending it nationwide.

The key, he said, is that each is grounded in the idea that gays and lesbians deserve equal protection for their relationships.

“To me that’s incredibly important,” Herrera said.

The court seemed to split along its traditional liberal/conservative lines, with Kennedy as a cautious pragmatist.

Chief Justice John Roberts wondered whether the fight over marriage was “just about the label,” because California grants to gay and lesbian couples all marital rights.

Roberts compared it to telling a child whom to call a friend. “I suppose you can force the child to say, this is my friend, but it changes the definition of what it means to be a friend,” Roberts said.

Olson shot back that “certain labels” have great meaning in the law and culture, such as the word “citizen.”

Sotomayor challenged Olson to explain how, if marriage is a fundamental right, states could ban polygamy or incest. Olson said polygamy and incest are individual conduct involving exploitation and abuse, whereas denying marriage to gays and lesbians is to deprive a class of people of equal rights.

Justice Elena Kagan challenged the notion of the Prop. 8 defense that the sole state interest in marriage is to promote “responsible procreation” among heterosexuals.

The court has scheduled arguments in the Defense of Marriage Act for today. That law bans federal recognition of same-sex marriages.