The Supreme Court’s rulings on same-sex marriage are barely a month old, but they have generated a wave of lawsuits and political battles as gay-rights supporters continue their quest to hoist the rainbow flag in every state.

In Missouri, the state Supreme Court this week said it wants to hear more legal arguments, in light of the high court’s Defense of Marriage Act (DOMA) ruling, in the case of a deceased state trooper whose surviving same-sex partner has been denied benefits.

New Mexico’s attorney general, a Democrat, said this week that he would not defend the state’s law banning same-sex unions. Five years ago, the state approved a constitutional amendment against same-sex marriage.

In Ohio, two residents are demanding that their marriage in Maryland be recognized by the state as one of the partners faces a potentially fatal illness.

The high court’s rulings striking down part of DOMA and effectively allowing California to resume same-sex marriages have shown little sign of ending the national debate. Same-sex marriage is still against the law in more than three-fifths of the states.

Although the high court struck down part of the federal law, the justices stopped well short of voiding state-level laws and constitutional statutes banning same-sex marriage across the country. Also left for the courts and state governments to decide is how same-sex marriages legal in one state will be treated in a state where such unions are expressly banned.

Activists on both sides say the battle — both legal and political — is far from over.

“Freedom to Marry’s goal is simple: Win marriage nationwide,” said Evan Wolfson, a longtime gay-rights activist and founder of the marriage advocacy group, which seeks legalization in a dozen or more states by 2016.

Traditional-values groups acknowledge that they are facing cultural headwinds, but they also insist the outcome is far from settled.

Most gay-marriage battles now will be in “more conservative states, where we fully expect to prevail,” Frank Schubert, political director of the National Organization for Marriage, said Wednesday.

“A couple of pitched battles remain — especially in Illinois,” he said, but “we are working hard with African-American pastors to preserve marriage” as well as working with allies to respond to all court battles and campaigns.

The Supreme Court’s twin rulings June 26 overturned a part of DOMA and effectively cleared the way for same-sex marriage — and the battle over such unions — to resume in California. The rulings’ effects on the 37 states that do not permit same-sex marriage now is being tested.

Ending a long losing streak at the ballot box prior to 2012, gay-rights allies are pressing their advantage amid rising public support and a cascade of favorable court rulings and political votes. This year alone, three state legislatures approved same-sex marriage and California became the 13th state to offer gay nuptials days after the Supreme Court issued its ruling against voter-approved Proposition 8.

Freedom to Marry has named Hawaii, Illinois, New Jersey and Oregon as its immediate targets. All of these states have civil union or domestic partnership laws that can convert to same-sex marriage. In states including Arizona, Colorado, Michigan, Nevada, New Mexico, North Carolina, Ohio, Pennsylvania and Virginia, activists are laying the groundwork to repeal amendments or laws prohibiting same-sex marriage.

Legal battles

The American Civil Liberties Union has filed a lawsuit in Pennsylvania, where the state’s Democratic attorney general is refusing to defend the state’s marriage law. The same scenario is playing out in New Mexico, where Attorney General Gary King, a Democrat, revealed that he would not defend the state’s constitutional provision against same-sex marriage in response to a lawsuit filed by two Santa Fe men who want to marry.

“New Mexico’s guarantee of equal protection to its citizens demands that same-sex couples be permitted to enjoy the benefits of marriage in the same way and to the same extent as other New Mexico citizens,” Mr. King said.

The federal target is still DOMA. Despite the Supreme Court ruling, the law still permits states to refuse to recognize same-sex marriages from other states.

House Republicans said recently that they were suspending efforts to apply DOMA in other federal programs, and some 200 members of Congress are trying to repeal DOMA entirely through the Respect for Marriage Act. House Republicans also quietly have dropped the funding request to underwrite the defense of the law in the courtroom in the aftermath of the Supreme Court decision.

DOMA, “though mortally wounded, is still on the books,” Mr. Wolfson said. Thanks to the June 26 ruling, however, he reported that, like other married gay couples, he can start the process to sponsor his noncitizen spouse for a green card.

Ohio’s special case

The Ohio case, which has attracted national attention, illustrated the complex legal terrain for same-sex marriage.

On Monday, U.S. District Judge Timothy Black in Cincinnati said John Arthur and James Obergefell could be declared married even though Ohio does not recognize same-sex marriage. After flying to Maryland this month to marry on the airport tarmac, the men asked the Ohio judge to recognize their marriage upon their return home.

The men’s lawsuit said Mr. Arthur is in poor health, and unless he is deemed legally married, his spouse will not be able to be buried with him in the Arthur family plot.

Judge Black ruled in their favor, saying Ohio recognizes other out-of-state marriages — such as those between first cousins — that aren’t permitted in the state. “How then can Ohio, especially given the historical status of Ohio law, single out same-sex marriages as ones it will not recognize,” Judge Black wrote. “The short answer is that Ohio cannot.”

Ohio backers of traditional marriage say the case — with a terminally ill plaintiff — was calculated to undermine the state’s law, with a sympathetic court aiding the process.

“Judge Black is an Obama appointee,” Ohio pro-life activist Pendra Lee Snyder, the owner of a Christian marketing and publishing company, told the website LifeSiteNews.com. “This is, in my opinion, a calculated effort at the chipping away of states’ rights and in particular all states that voted to protect marriage as a union of one and one woman.”

Fixing for a fight

Supporters of traditional marriage are regrouping and resisting adverse decisions.

In Congress, 40 members of the House, led by Rep. Tim Huelskamp, Kansas Republican, have introduced a constitutional amendment to define marriage as the union of one man and one woman.

Conservatives also are rallying research and rhetoric to support their position, as well as cataloging cultural skirmishes such as photographers and bakers who are being penalized for not assisting with same-sex weddings.

“No American — business owner or not — should have to violate their beliefs to compete in the marketplace,” said the Family Research Council, noting that 85 percent of 1,000 Americans in a Rasmussen Reports poll said a Christian photographer who opposes same-sex marriage on religious grounds shouldn’t be forced to work at a wedding between gays.

• This article is based in part on wire service reports.

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