GOP congressional leaders, unable to roll back the law while President Obama remains in office and unwilling to again threaten a government shutdown to pressure him, are focused on other issues, including trade and tax reform.
Less noted, senior Republican lawmakers have quietly incorporated many of the law's key protections into their own proposals, including guaranteeing coverage and providing government assistance to help consumers purchase insurance.
And although the law remains very unpopular with GOP voters, more than 20 million Americans now depend on it for health benefits, making even some of the most conservative Republicans loath to cut off coverage.
Facing the prospect that the Supreme Court this year could strip away insurance subsidies provided through the law in most states, several GOP lawmakers have proposed extending the aid, perhaps even until a new president takes office.
At the same time, the presumed Republican presidential front-runner, former Florida Gov. Jeb Bush, has shown little enthusiasm for a new healthcare fight. Last year, he even criticized the repeal effort.
Republicans who still demand a repeal, including Sen. Ted Cruz of Texas and Louisiana Gov. Bobby Jindal, appear to be long shots for the presidential nomination.
"Only 18% of Americans want to go back to the system we had before because they do not want to go back to some of the problems we had," Whit Ayres, a veteran Republican pollster who works for presidential candidate Sen. Marco Rubio of Florida, said at a recent breakfast hosted by the Christian Science Monitor.
"Smart Republicans in this area get that," he added.
These developments have sapped enthusiasm among Republican leaders for "pulling out Obamacare root and branch," as Senate Majority Leader Mitch McConnell (R-Ky.) once put it.
The waning of the repeal campaign does not mean that calls to undo the law will vanish from the race for the 2016 Republican presidential nomination.
Nor is the law likely to remain unchanged, particularly if millions of people lose coverage as a result of a Supreme Court ruling.
The unpopular mandate that Americans have insurance could be adjusted, as could some taxes and other coverage requirements.
A little-noted provision in the law that allows states to develop alternative systems for coverage could someday produce substantial local variations.
But the GOP confronts a very different landscape from 2010, when the party pledged full repeal with warnings the law would "ruin the best healthcare system in the world and bankrupt our country," as then-House Minority Leader John A. Boehner (R-Ohio) said.
Republicans' hopes that the Supreme Court would rule the law unconstitutional were dashed in 2012, when the justices decided otherwise. The GOP missed another opportunity when Obama won reelection later that year.
In 2013, congressional Republicans' maneuver to shut down the government became a debacle, prompting McConnell to promise no more shutdowns over the healthcare law.
Today, more than 11 million Americans, many of them previously uninsured, get health coverage through marketplaces created by the law.
Another 11 million low-income people have signed up for Medicaid, mostly in states that expanded their programs with federal funding made available through the law.
That has fueled a historic coverage expansion. In the first quarter of this year, 11.9% of adults in the U.S. lacked insurance, down from 18% in the third quarter of 2013, before the current expansion began, according to Gallup.
Ripping out this new system — and jeopardizing coverage for millions — would present enormous political and practical challenges.
"One key lesson of Obamacare has been that people are not eager for disruption," said Yuval Levin, a leading conservative healthcare expert and former advisor to President George W. Bush.
Republican lawmakers once cheered a legal challenge targeting insurance subsidies in about three dozen states as an opportunity to dismantle the law.
Now, they are scrambling for a plan to preserve the subsidies if the Supreme Court backs the challenge. A ruling in the King vs. Burwell case is expected in June. Sen. Ben Sasse (R-Neb.) has proposed legislation to extend the aid for 18 months. Sasse and other Republican lawmakers say the time is needed to develop a replacement for the current law.
But even the process of developing a replacement plan is shaping up to produce something less groundbreaking than Republicans once promised.
No longer is the party proposing to overhaul the employer-based system that most Americans rely on for health benefits, as Sen. John McCain (R-Ariz.) did when he was the presidential nominee in 2008.
Nor are party leaders pushing plans that don't meaningfully expand coverage, as House Republicans did in 2009.
In fact, the most fully developed GOP healthcare plan at the moment not only preserves the employer system, it essentially keeps many provisions of the Affordable Care Act, though sometimes in a different form.
The proposal — by Sen. Richard M. Burr (R-N.C.), Senate Finance Committee Chairman Orrin G. Hatch (R-Utah) and House Energy and Commerce Committee Chairman Fred Upton (R-Mich.) — would guarantee coverage to Americans even if they are sick, a protection that was a watershed when it was enacted in the 2010 law.
The Republican outline would ban insurance companies from imposing lifetime limits on coverage, allow young adults up to age 26 to remain on their parents' health plans, and create a system of tax subsidies to help Americans buy coverage. All are pillars of the current law.
Like the current law, the proposal includes a system for penalizing people who don't have health insurance, although the mechanism is different than the current tax penalty.
And the Republican plan, like the current law, relies on cuts in Medicare spending and a new tax on employer-provided health plans with particularly rich benefits.
"This acknowledges that the ACA is the law and … you have to start with what is there and build on it," said former Utah Gov. Mike Leavitt, who served as Health and Human Services secretary in the last Bush administration. "The recipe is to begin pushing it philosophically to the right."
To be sure, the debate over how to do that would probably still be difficult and contentious.
Republicans have proposed changing the government Medicaid insurance program for the poor into a system of block grants to states. That provokes fierce opposition from Democrats.
GOP calls to eliminate federal requirements that insurers cover a basic set of benefits threaten what Democrats consider a key protection in the law. Republicans say this regulation should be left to states.
But in contrast to the unprecedented repeal campaign, such debates would reflect a more established tradition of adjusting existing law.
"This is generally how public policy evolves in this country," said Brookings Institution senior fellow Stuart Butler, a veteran conservative policy expert who worked with Republicans two decades ago to reshape the federal welfare system. "You don't start with a completely blank slate."