President Donald Trump has said the new Republican healthcare policy should be to allow the current law to collapse.
"I'm not going to own it," Mr Trump told reporters of Obamacare, "I can tell you the Republicans are not going to own it."
Support for the Republican Senate bill fell apart on Monday when two more senators said they could not back it.
Senate Democratic leader Chuck Schumer said Mr Trump was "playing a dangerous game" with the US healthcare system.
"He is actively, actively trying to undermine the healthcare system in this country using millions of Americans as political pawns in a cynical game", Mr Schumer said on Tuesday.
Republican Senate leader Mitch McConnell said the chamber would vote early next week on a motion for repealing Obamacare only.
But with at least three Republicans against the plan B, it is probably doomed, too.
Not a good strategy - Anthony Zurcher, BBC News, Washington
Call it Mr Trump's "Joker" option for Obamacare repeal. Do nothing, and watch the world burn.
Explaining how your dastardly plan will go down seldom works for cartoon villains, however, and it may not be the best strategy for presidents either.
By telling Americans he's going to let the US healthcare system collapse in order to rally support for an eventual fix, he's essentially taking responsibility for anything bad that happens in the coming days. Despite his protestations that he won't "own" what happens next, he just stuffed the receipt in his pocket.
Premiums go up? Insurance providers pull out of state individual markets? Democrats can say it's part of the president's plan to sabotage the system.
All the while Mr Trump's opponents can point to the Congressional Budget Office's findings that, without White House meddling, the insurance markets would be relatively stable.
Then again, at this point, the president has few good options. His promises to eventually work with Democrats are undermined by his pledges to defeat them at the ballot box next year. He clearly can't move forward with Republicans alone.
All that's left is to walk away and hope for the worst.
President Trump backed the just-repeal-it plan on Monday night but changed his tune on Tuesday morning by proposing to simply let Obamacare - which has failed to curb rising costs as insurance options dwindle - die on its own.
"As I have always said, let ObamaCare fail and then come together and do a great healthcare plan. Stay tuned!" he tweeted.
He has invited all Republican senators to discuss healthcare over lunch at the White House on Wednesday.
Without a replacement bill, analysts have estimated that millions of people would lose health insurance.
What had Republicans proposed?
The party's proposed alternative includes steep cuts to Medicaid, a healthcare programme for the poor and disabled, removed the individual mandate requiring all Americans to have health insurance or pay a tax penalty and implemented a six-month lockout period for anyone who lets their health coverage lapse for more than two months.
The House of Representatives passed a similar version of the Senate bill, but slashed taxes on the wealthy used to pay for the health scheme. The Senate proposed a similar provision but was forced to ditch it amid opposition.
Why did it fail?
Two Republican senators, Mike Lee and Jerry Moran, came out against the bill on Monday night, saying the new legislation did not go far enough in repealing Obamacare.
They joined Senators Rand Paul and Susan Collins, who had already opposed the bill.
Republicans, who hold 52 seats, no longer have enough votes to approve the measure in the 100-member Senate.
Moderate Republicans had also said the bill would have harmed some of their vulnerable constituents.
The non-partisan Congressional Budgetary Office (CBO) found the bill would have stripped 22 million Americans of health insurance over the next decade.
On top of that, polls show Obamacare has gained popularity among Americans - a survey by the Washington Post and ABC News on Monday found more than twice as many people preferred Mr Obama's programme to the proposed alternative.
It is estimated Obamacare extended insurance to 20 million more Americans.
What happens now?
Mr McConnell had hoped to push through a 2015 plan to repeal Obamacare only and give lawmakers a "two-year, stable transition" period to give Congress time to come up with a replacement plan.
But Susan Collins, Shelley Moore Capito and Lisa Murkowski have said they will not support their party's proposal to repeal Obamacare without a replacement plan.
All but two Senate Republicans, including Mrs Collins, voted in favour of the repeal plan two years ago.
"We should not give up," Mr McConnell said on the Senate floor Tuesday.
"Passing this legislation will provide an opportunity for senators of all parties to engage with a fresh start and a new beginning for the American people."
Earlier this month, Mr McConnell had said: "I think repealing and then delaying the replacement doesn't work".