Washington marched relentlessly toward its first federal shutdown in 17 years Saturday after House Speaker John A. Boehner yielded to conservative demands to use a government-funding bill to press an attack on President Obama’s 2010 health-care law.

Early Sunday morning, the House approved a plan to keep the government open past midnight Monday. But under pressure from Sen. Ted Cruz (Tex.) and other GOP hard-liners, Boehner (R-Ohio) agreed to amend the bill to include a one-year delay of the health law’s mandates, taxes and benefits — ensuring a stalemate with the Democratic Senate.

The amendment passed on a largely party-line vote, 231 to 192, with two lawmakers from each party breaking ranks to vote with the other side.

“We will do everything we can to protect Americans against the harmful effects of Obamacare. This bill does that. We’re united in the House as Republicans,” Majority Leader Eric Cantor (Va.) said Saturday night shortly after unveiling the plan to his rank and file. “Now it’s up to the Senate Democrats to answer.”

That response came quickly. Even before the House had a chance to vote on its proposal, Senate Majority Leader Harry M. Reid (D-Nev.) blasted it as “pointless.” Democratic aides said the Senate would set aside the House amendments, probably first thing Monday, leaving GOP leaders with a stark choice: approve the simple funding bill the Senate has already passed or permit federal agencies to close.

Countdown to the (possible) shutdown

“As I have said repeatedly, the Senate will reject any Republican attempt to force changes to the Affordable Care Act through a mandatory government funding bill,” Reid said in a written statement. “After weeks of futile political games from Republicans, we are still at square one.”

White House press secretary Jay Carney called the latest GOP strategy “reckless and irresponsible.”

“Any member of the Republican Party who votes for this bill is voting for a shutdown,” Carney said in a written statement.

As the House convened for the rare Saturday session, senior Republicans seemed to recognize the potential consequences of their actions. For now, Boehner’s decision to appease his right wing keeps an uneasy peace in his fractious caucus. But it bodes ill for his ability to work with Democrats to keep the government open, restore funding for federal agencies if a shutdown occurs or — in a few weeks — raise the federal debt limit to avoid a first-ever default on the national debt.

Leaders of both parties agree that a government shutdown would be bad for the economy and that a default would be potentially catastrophic. Complicating hopes for an easy resolution: Obama leaves Saturday for a week-long trip to Asia. Meanwhile, the maneuvering of House Republicans has caused considerable anxiety within their party.

“I think it’s going to be tough for them. They’re having such difficulty pulling things together,” said Sen. Bob Corker (Tenn.), one of several GOP senators who consults frequently with House members. “I don’t know that I have a clear vision how we move through this. And I think the debt ceiling is maybe even more murky.”

As recently as late August, Boehner urged his rank and file to avoid provoking a fight that could shut down the government, arguing that Republicans would have more leverage during the battle over the $16.7 trillion debt limit. Treasury Secretary Jack Lew has said Congress must raise the debt limit by Oct. 17 or the nation will begin running short of cash to pay its bills.

Many Democrats view a government shutdown as unpleasant but economically survivable, and polls suggest that voters would blame Republicans. But raising the debt limit is an entirely different matter, a deadline Obama and other Democrats are unwilling to breach.

“Failure to meet this responsibility would be far more dangerous than a government shutdown. It would effectively be an economic shutdown, with impacts not just here, but around the world,” Obama said Friday at the White House. “We don’t fully understand what might happen, the dangers involved.”

But over the summer, outside groups such as Heritage Action for America kicked up dust about the health law, which is set to begin signing up consumers Tuesday. Cruz and other far-right Republicans refused to pass up a single opportunity to attack it, arguing that the law would ravage the economy, destroy jobs and raise insurance premiums for average Americans.

“There’s a real push to say we’re going to do whatever we can, as much as we can, to protect the people of our districts from the harmful effects of this law,” said Rep. James Lankford (R-Okla.), a member of the conservative Class of 2010 who now serves in House leadership. “So let’s find out how much we can protect them from.”

That summer onslaught set in motion a series of capitulations by Boehner. First, he agreed to include a provision to defund the health law in the government-funding measure the House initially sent the Senate. Despite a 21-hour talkathon, Cruz and his allies failed to force the Senate to uphold that version.

Boehner then tried to declare a temporary cease-fire in the health-care fight and once again urged conservatives to defer their battle to the debt-limit debate. But conservatives again objected, forcing Boehner on Saturday to agree to advance a one-year delay of the health law as part of the government-funding bill, formally known as a “continuing resolution,” or C.R.

In addition to delaying implementation of the law, the amendment would weaken its requirement that insurance policies cover contraception with no co-payments, instead allowing employers to decide whether to offer such coverage to their workers.

That drew fire from birth-control advocates, such as Dawn Laguens, executive vice president of the Planned Parenthood Federation of America, who called it “a remarkably desperate, misguided and extreme attack on women’s health.”

The latest House strategy also includes a proposal to repeal a tax on medical devices that helps fund the health-care law and has long been unpopular in both parties. The House voted 248 to 174 to approve that change.

And Democrats joined Republicans to overwhelmingly approve a separate measure that would guarantee that active-duty members of the military — as well as civilians and contractors essential to their work — would get paid in the event of a shutdown. That would eliminate one of the most politically sensitive consequences for Republicans if a shutdown occurs.

But Democratic aides said the Senate was unlikely to consider that bill, either.

“We are sending them a bill to pay our troops. It’s called the C.R.,” Reid spokesman Adam Jentleson said via e-mail.

On Saturday afternoon, senior GOP aides in the House held out hope that the Senate might feel obliged to approve the repeal of the medical-device tax. Earlier this year, the Senate voted overwhelmingly to repeal and replace the tax.

But Reid quickly shut down such hopes. Senate aides in both parties confirmed that Reid would need only 51 votes to table both of the House amendments, denying Cruz and his allies the opportunity to block Democrats and keep the House’s changes alive.

Some rank-and-file House Republicans said they now see no way to avoid a shutdown. Rep. Thomas J. Rooney (R-Fla.) called the prospect “likely.”

For now, at least, Boehner, Cantor and other top lieutenants have ruled out the prospect of seeking out Democratic votes to help them pass a simple funding bill and keep the government open, their advisers said. But as Republicans prepared to vote late Saturday, some lawmakers acknowledged that they had no idea what would happen if the Senate follows through on its threat to reject their latest offering.

“It comes back to us, I guess,” said Rep. Phil Gingrey (Ga.), one of the more conservative Republicans and a candidate for Senate in 2014. “We really didn’t talk about exactly what the plan would be then.”

Ed O’Keefe, Jeff Simon, Jackie Kucinich and David A. Fahrenthold contributed to this report.