Six years after President Obama rammed Obamacare through a Democrat-run Congress, Republicans used their own majority to take a “monumental” step toward scrapping the overhaul Wednesday, approving a bill to ax the onerous mandates that penalize Americans who refuse to comply with the health care law’s demands.

After dozens of false starts, Republicans managed to get a bill through both chambers of Congress and sent the first big repeal to Mr. Obama, who will veto it.

There is little chance of Republicans overriding the veto, but the highlight-reel moment is designed to rally conservatives and set the tone for the election year.

House Speaker Paul D. Ryan, Wisconsin Republican, is positioning his chamber as a factory for ideas that offer a clear contrast to the Obama era.

“The people deserve a truly patient-centered health care system — and ultimately, this is going to require a Republican president,” Mr. Ryan said. “That’s why our top priority in 2016 is going to be offering the country a clear choice with a bold pro-growth agenda.”

The House approved the bill late Wednesday on a 240-181 vote near party lines. The Senate honed and approved the package on a 52-47 vote in early December.

Three Republicans voted “no,” and one Democrat — Rep. Collin C. Peterson of Minnesota — supported the measure, which used a fast-track budget process to bypass a filibuster in the Senate.

Each of the Republicans who bucked the party — Reps. Robert J. Dold of Illinois and John Katko and Richard L. Hanna of New York — is a centrist.

Mr. Dold and Mr. Hanna voted against defunding Planned Parenthood in September.

The bill that passed Wednesday strips federal funding from Planned Parenthood for one year. Republicans sought to punish the organization after undercover videos showed officials negotiating sales of fetal body parts. In two weeks, thousands of marchers are expected to descend on Washington to protest abortion in the annual March for Life.

Though Republicans have sought to repeal the Affordable Care Act from the moment Mr. Obama signed it in March 2010 and the House has approved repeated repeals over the past five years, this legislation is the first to clear the Senate as well.

The 2014 elections gave Republicans a majority in the upper chamber to go along with their House majority, and they turned to the same fast-track budget tool Senate Democrats used in 2010 to pass Obamacare.

Known as “reconciliation,” the process allowed Republicans to pass revenue-related legislation on a majority vote in the Senate, sidestepping the threat of a Democratic filibuster. Congressional scorekeepers said the bill would save taxpayers a half-trillion dollars over the next decade.

The experience is supposed to serve as a road map for killing the law next year if Republicans capture the White House in November and hold their majorities in both houses of Congress.

For now, Mr. Ryan and his chief lieutenants are daring Mr. Obama to take political accountability for his health care law, as customers in some parts of the country face the “sticker shock” of rising premiums even as they are mandated to hold insurance.

The nation’s largest insurer, UnitedHealth Group, recently said it is losing money on the health care exchange and may pull out entirely by next year, and more than half of Obamacare’s 23 nonprofit co-ops stopped offering plans this year, fueling Republicans’ arguments.

“We are confronting the president with the hard, honest truth — Obamacare doesn’t work,” Mr. Ryan said at a press briefing Wednesday.

Democrats say the repeal is ill-fated and misguided. The bill would strip Obamacare-derived coverage from 22 million Americans, they said, including poor residents in 30 states that expanded Medicaid.

Ten of those states are represented by Senate Republicans up for re-election in November, and Democrats predict voters will retaliate against them.

The Republican-authored bill also would phase out taxpayer subsidies for people who shopped for coverage on Web-based Obamacare exchanges. Open enrollment on the portals lasts until Jan. 31, and the administration said Wednesday that roughly 8.6 million people have selected or renewed plans for this year on the federal HealthCare.gov website, which serves 38 states.

“We are in the final weeks of open enrollment for 2016 under the Affordable Care Act and saw unprecedented demand for January 2016 coverage. Yet at the same time, we continue to see efforts to repeal the ACA and turn back the clock,” Health and Human Services Secretary Sylvia Mathews Burwell said after the vote.

Among other provisions, the bill would scrap the “individual mandate” that requires most Americans to hold insurance — a lever designed to drive healthy customers into the marketplace to make Obamacare’s fragile economics work — and the government could no longer force large employers to provide health care coverage or pay hefty fines.

Republicans haven’t settled on a long-overdue plan to replace Obamacare, and Democrats needled them for trying to tear down Mr. Obama’s vision without posting an alternative.

“Republicans have been full of talk, and we haven’t seen any action,” said Rep. Chris Van Hollen, Maryland Democrat.

Democrats also said Republicans were setting themselves up for failure in the fall elections. They cast the Republican-authored bill as a “one-two punch” against female voters who rely on preventive health care services covered by Obamacare and offered by Planned Parenthood clinics across the country.

“Here we are, Week One of 2016, and we have a multifaceted attack on women’s health,” said Rep. Jared Polis, Colorado Democrat.

Republicans have been sparring with Planned Parenthood since taking control of the House in 2011 and attaching language defunding the organization to the very first spending bill.

That fight re-emerged last year when pro-life activists released a series of videos suggesting that the organization was skirting the law by negotiating the sale of tissue taken from aborted fetuses. Such sales are legal only if they are not for profit.

Planned Parenthood says it has done nothing wrong and has welcomed scrutiny from the inspector general’s office of the Department of Health and Human Services.

But Republican leaders have argued that taxpayers should not be subsidizing organizations that perform abortions and harvest fetal tissue, even though clinics and other health care providers are barred from spending federal funds on the procedure itself. Their plan is to redirect $235 million of the organization’s funding to women’s health clinics that do not offer abortion.

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