President Trump threatened Tuesday to let the Obamacare law collapse under its own weight and then urged Senate Republicans to eliminate legislative filibusters as a way to advance his agenda after their replacement health care bill derailed.

After briefly backing the plan of Senate Republican leadership to pass a repeal of Obamacare and later craft a replacement, Mr. Trump settled on the let-it-die option that he said would force Democrats to negotiate a new health care law.

“It will be a lot easier, and I think we’re probably in that position where we’ll let Obamacare fail,” the president told reporters at a White House lunch with troops from the Afghanistan War.

“We’re not going to own it. I’m not going to own it. I can tell you the Republicans are not going to own it,” he said. “We’ll let Obamacare fail, and then the Democrats are going to come to us.”

The stark pronouncement underscored the painful defeat for Mr. Trump, with Senate Republicans unable to hold together a thin 52-48 majority to keep their 7-year-old promise to repeal Obamacare.

The Senate repeal-and-replace bill collapsed after two more Republicans came out against it Monday, bringing total GOP opposition to four.

Wall Street was largely unfazed by the drama on Capitol Hill. The Dow Jones industrial average closed down, but the tech-heavy Nasdaq composite index soared to a record high on a Netflix rally.

Analysts said the expectation that Mr. Trump will succeed in cutting taxes outweighed concerns about health care laws and the problems with Obamacare still could be worked out.

“Despite recent legislative setbacks, we continue to believe that a tax bill is more likely than not to become law in 2018, though there remain many unanswered questions, including the potential size. We expect a $1 trillion tax cut over 10 years,” said Goldman Sachs economist Alec Phillips.

Mr. Trump’s approach to the legislative process has been all over the map.

He endorsed the “repeal-then-replace” idea in recent weeks and again on Twitter late Monday before assigning blame the next morning.

“We were let down by all of the Democrats and a few Republicans. Most Republicans were loyal, terrific & worked really hard. We will return!” he said during a morning tweet storm.

Late Monday, he said Republicans should deliver a repeal bill and, with the Democrats’ signature law gutted, pressure the minority party to join the replacement process.

On Tuesday, he said Obamacare should wither on the vine.

“As I have always said, let ObamaCare fail and then come together and do a great healthcare plan. Stay tuned!” he tweeted.

Still, the let-it-die plan offered by Mr. Trump is fraught with political pitfalls, including the perception that the president would stand idly by while Americans lose health insurance or suffer skyrocketing premiums.

Senate Minority Leader Charles E. Schumer, New York Democrat, seized the opportunity to accuse the president of “actively trying to undermine the health care system in the country [and] using millions of Americans as political pawns in a cynical game.”

He said Senate Republicans had to choose between the president’s “sabotage” of the health care system or joining Democrats in a bipartisan effort to fix the ailing Obamacare law.

White House deputy press secretary Sarah Huckabee Sanders insisted that congressional Democrats were responsible for the fate of the health care bill that they wrote and passed with no Republican support in 2010.

“They created the program. They pushed it through. They made this legislation happen, and they need to own the failure of it. I think they’ve been the ones that have been completely unwilling to even come to the table to be part of the discussion,” she said at the daily White House press briefing.

She acknowledged that Mr. Trump was frustrated by the Republican bill’s collapse.

“I think his primary frustration is that there is no progress in terms of over the last 24 hours of moving this further down the road and giving Americans the system that they deserve,” she said. “But I think in large part, most of the frustration lies with the Democrats, who created the mess but don’t want to help fix the problem.”

Without moves to prop it up, the Obamacare system is collapsing. Insurers are pulling out of government-run exchanges and rapidly increasing premiums.

Senate Majority Leader Mitch McConnell, Kentucky Republican, announced that the Senate would take up a repeal bill passed in 2015 that would be phased in over two years, providing time to enact a replacement.

However, at least three Republican senators immediately announced opposition to the bill. That was enough to doom it.

Mr. Trump also urged Senate Republicans to change the chamber’s rules to eliminate legislative filibusters, saying it was the only way to break Democratic obstruction and move the country forward.

“With only a very small majority, the Republicans in the House & Senate need more victories next year since Dems totally obstruct, no votes!” Mr. Trump said in a Twitter post. “The Senate must go to a 51 vote majority instead of current 60 votes. Even parts of full Repeal need 60. 8 Dems control Senate. Crazy!”

The move faces near universal opposition in the Senate because it would dramatically change the deliberative nature of the upper chamber. The end of the filibuster would transform the Senate into a majority-rule institution resembling the House.

The filibuster for most executive appointments was nixed when the Democratic majority in 2014 used the “nuclear option” to push through President Obama’s judicial nominees.

Republicans this year ended the filibuster of Supreme Court nominees to put Neil Gorsuch on the high court.

Mr. Trump has previously called for an end to the Senate filibuster to help advance his stalled agenda. He has not scored a major legislative victory as he approaches the six-month mark since taking office.

• Dave Boyer and Tom Howell Jr. contributed to this report.

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