WASHINGTON — The Supreme Court’s resounding rejection of a conservative attempt to gut President Barack Obama’s health care overhaul won’t stop Republicans from attacking the law they detest. But now, their efforts will be chiefly about teeing up the issue for the 2016 presidential and congressional elections.

The court’s decision left GOP lawmakers stunned and uncertain about some of their next steps. Most agreed they would continue trying to annul the entire law and erase individual pieces of it, like its taxes on medical devices. Yet many also conceded they have little leverage to force Obama to scale back — let alone kill — one of his most treasured legislative achievements.

“Obamacare is fundamentally broken,” said House Speaker John Boehner, R-Ohio, who said repeal efforts would continue. But he declined to guarantee a vote on a replacement plan this year and said, “It’s very difficult to deal with it when you have a president that fundamentally disagrees with us. And so the struggle will continue.”

“We’ll continue to pick away at the law,” said Sen. John Barrasso, R-Wyo., a leader of Senate GOP efforts to craft a bill responding to a court victory that never came. “But ultimately our goal is to repeal and replace, and that’s not going to be possible until after the 2016 elections,” when Republicans hope for a GOP president.

That leaves Republicans mostly using efforts to void or dramatically weaken the health care overhaul to contrast their views with Obama’s for voters.

“It’s going to be one of the most important, if not the most important, debating points for 2016,” said Rep. John Fleming, R-La.

Yet though few would admit it, the court’s ruling caused many Republicans to exhale with relief.

By leaving intact federal subsidies that help millions afford health care, the ruling lets Republicans bash Obama’s law in their 2016 campaigns without having to quickly help those who would have lost assistance. Taking away those benefits could have created many angry voters.

“There isn’t some sort of, that sword of Damocles hanging over our heads that people are going to lose their care in 30 days or something,” said Rep. Greg Walden, R-Ore., who heads House GOP campaign efforts.

Republicans have promised to repeal and replace Obama’s law since its 2010 enactment, but have yet to rally behind a plan doing that. A bill temporarily continuing the subsidies this year would have faced a difficult path to passage, thanks to conservative reluctance to abet Obama’s health law in any way.

Had the GOP-backed lawsuit by conservatives succeeded, the 34 states likeliest to lose subsidies included many that will be pivotal in next year’s elections. That included Florida, a key presidential state where a national high of 1.3 million people could have lost assistance, plus Pennsylvania, Illinois, Ohio, Wisconsin and New Hampshire, where GOP senators could face tight re-election battles and worried about constituents losing aid.

“The court has saved Republicans from themselves,” said former Rep. Tom Davis of Virginia, who once led the House Republican political organization.

Underscoring that the court decision left intact the issue’s high-wattage political appeal, it took just minutes for both sides to use the ruling for political fundraising. GOP hopeful Jeb Bush, the former Florida governor, emailed for money to stop “the overreaches of the last six years,” while Senate Democrats warned that Republicans have “pledged to destroy Obamacare.”