WASHINGTON

President Obama exhorted fellow Democrats on Wednesday to preserve his legacy-defining health care law as Republicans moved ahead with their long-desired bid to scrap it in what Vice President-elect Mike Pence called the “first order of business” of the incoming Donald Trump administration.

The Republican-controlled U.S. Senate brushed aside unified opposition by Democrats and voted to open debate on a resolution setting in motion the Republican drive to repeal the 2010 Affordable Care Act, which has helped upwards of 20 million previously uninsured Americans obtain health insurance.

As early as 2018, the millions of people who gained insurance under the law could see their coverage in jeopardy — especially if Congress fails to find a replacement to the law beforehand.

President Obama, who hands over the presidency to Mr. Trump on Friday, Jan. 20, made a rare trip to Capitol Hill to urge Democratic lawmakers to protect the measure, which informally is known as Obamacare and is considered his signature domestic policy accomplishment.

Republicans, who will control the White House as well as both chambers of Congress when Mr. Trump takes office, stepped up their rhetorical attack on the law, which they have labeled a government overreach.

Democrats in turn accused them of trying to rip apart the nation’s health care system with no firm plan to replace it.

“The Republican plan to cut health care wouldn’t ‘make America great again,’” Senate Democratic Leader Chuck Schumer told reporters, invoking the Trump campaign slogan.

“It would make America sick again and lead to chaos instead of affordable care.”

Since the law was enacted, Republicans in Congress have voted more than 50 times to try to repeal all or part of it and conservatives have filed suits to try to invalidate it.

Republicans, who have fought nearly all of President Obama’s major legislative initiatives during his eight years in office, have said Obamacare has brought excessive government intrusion into the healthcare market and contend it is harming job growth by adding burdens on businesses.

President Obama “encouraged us to fight,” Maryland Democratic Rep. Elijah Cummings told reporters after meeting with the president. President Obama told reporters his message was, “Look out for the American people.”

Mr. Pence, the outgoing Indiana governor and a former member of the U.S. House of Representatives, met Republican lawmakers to plot the path forward on scuttling the law.

Afterward, they stepped up their rhetorical attack on Obamacare with House Speaker Paul Ryan of Wisconsin saying the law ruined the American health care system.

“The first order of business is to keep our promise to repeal Obamacare and replace it with the kind of health care reform that will lower the cost of health insurance without growing the size of government,” Mr. Pence said at a news conference.

Mr. Pence said Mr. Trump will work in concert with congressional leaders for a “smooth transition to a market-based health care reform system” through legislative and executive action.

During two news conferences, Mr. Pence, Speaker Ryan and Senate Majority Leader Mitch McConnell of Kentucky offered few details on what a Republican-backed replacement for Obamacare would look like.

Speaker Ryan said lawmakers will take action that does not “pull the rug out from anybody” and that Republicans “have a plan” and “plenty of ideas.”

Democrats are trying to depict Republicans as risking breaking the entire health care system — including for those who currently get their insurance through an employer and not under provisions created by the ACA.

Republicans are “pulling the string that’s going to unravel the whole (healthcare) system,” Michigan Democratic Sen. Debbie Stabenow said.

Mr. Trump wrote earlier on Twitter that Republicans “must be careful in that the Dems own the failed ObamaCare disaster, with its poor coverage and massive premium increases.”

“Don’t let the Schumer clowns out of this web,” Mr. Trump added.

Sen. Schumer of New York said Democrats would not take any blame: “They want to repeal it and then try to hang it on us. Not going to happen. It’s their responsibility, plain and simple.”

The Republicans risk causing turmoil in the health insurance market and generating a political backlash if their policies fail. Mr. Pence said repealing Obamacare must be done in a way that does not inflict hardship on the economy or on Americans who obtained insurance through the law.

“The architecture of the replacement of Obamacare will come together, as it should, through the legislative process in the weeks and months ahead,” Mr. Pence said.

House Republicans last year offered a proposal that would, among other things, provide refundable tax credits to help people afford their medical insurance premiums.