WASHINGTON (AP) — It was a day of history for the nation — and sweet vindication for President Obama. His grin seemed wider than any in recent memory.

After more than a year of arguing, struggling and dealing, Obama signed into law a nearly $1 trillion health care overhaul that ranks among the biggest changes ever devised by Washington and will reshape the way virtually every American receives and pays for treatment. It will rework fully one-sixth of the U.S. economy and for the first time cement insurance coverage as the right of every U.S. citizen.

At the White House, jubilation was in the air on Tuesday. Democratic lawmakers and advocates, crowded into the East Room for the signing ceremony, hooted and hollered at nearly every Obama sentence. They snapped photos of the president — and themselves. Vice President Joe Biden was caught whispering a profanity as he exclaimed to the president what a big deal it was.

It seemed more like a campaign rally than a bill signing. No one seemed quite able to contain themselves.

It will be months before the November elections render a verdict on whether the public approves of the bill the Democrats pushed through Congress without a single Republican vote. But after more than a year of hyperpartisan struggle — and numerous near-death moments for the measure — they sealed a victory denied to a line of presidents and Congresses stretching back more than a half-century.

Not everyone was cheering. Republicans said those Democratic lawmakers would pay dearly in this November’s elections. Opinion polls show the public remains skeptical, too, and Obama will fly to Iowa on Thursday for the first of a number of appearances that will be more like a continuing sales job than a victory lap.

Aside from the huge, real-life changes in store for many Americans, the White House hopes the victory — even as a companion Senate ‘‘fix-it’’ bill moves through the Senate — will revitalize an Obama presidency that has been all but preoccupied with health care for his first year and two months in office.

Republicans characterize the measure as a costly, wrongheaded government power grab. Obama and the Democrats portray it as literally a lifesaver for countless Americans.

The core of the massive law is the extension of health care coverage to 32 million who now lack it, a goal to be achieved through a complex cocktail of new mandates for individuals and employers, subsidies for people who can’t afford to buy coverage on their own, consumer-friendly rules clamped on insurers, tax breaks, and marketplaces to shop for health plans.

The law’s most far-reaching changes don’t kick until 2014, including a requirement that most Americans carry health insurance — whether through an employer, a government program or their own purchase — or pay a fine. To make that a reality, tax credits to help cover the cost of premiums will start flowing to middle-class families and Medicaid will be expanded to cover more low-income people.

Among the new rules on insurance companies are banning lifetime dollar limits on policies, coverage denials for pre-existing conditions, and policy cancellations when someone gets sick. Insurers also will have to allow parents to keep children on their plans up to age 26.

The changes are to be paid for with cuts in projected government payment increases to hospitals, insurance companies and others under Medicare and other health programs, an increase in the Medicare payroll tax for some, fees on insurance companies, drug makers and medical device manufacturers, a new excise tax on high-value insurance plans and a tax on indoor tanning services.

For seniors, the plan will gradually close the ‘‘doughnut hole’’ prescription coverage gap and improve preventive care. But it also will cut funding for popular private insurance plans offered through Medicare Advantage. About one-quarter of seniors have signed up for such plans, which generally offer lower out-of-pocket costs.

Democrats, led by Obama, celebrated a ‘‘new wind at our backs’’ from an achievement accomplished after more than a year of high tension and deep division — stretching back to shouted protests that interrupted lawmakers’ town hall meetings on the subject last summer. Obama signed the measure less than two days after the cliffhanger final House vote in a rare Sunday night session.

‘‘Our presence here today is remarkable and improbable,’’ Obama said. ‘‘With all the punditry, all of the lobbying, all the game-playing that passes for governing in Washington, it’s been easy at times to doubt our ability to do such a big thing, such a complicated thing.’’

At a second celebration later, he said, ‘‘After a century of striving, after a year of debate, after a historic vote, health care reform is no longer an unmet promise. It is the law of the land.’’

The president now faces the task of selling to the public a bill that satisfies neither side of the political spectrum.

Liberals bemoan that a government-run plan to compete with private ones was shed from the legislation during bitter negotiations. Conservatives fear an expansion of government and costs they say will bankrupt the country, despite an estimate from the nonpartisan Congressional Budget Office that the law will cut federal budget deficits by an estimated $143 billion over a decade.

Obama’s explanatory hurdle is not an easy one, given the law’s multilayered provisions and timetables. A bumper-sticker slogan it is not. But he must help protect the Democrats — particularly those from conservative-leaning districts — who stand to suffer in the fall elections from their votes.

Republicans face a challenge as well. Aware of traditional American suspicions of government intrusion, they cast themselves throughout the process as against major changes. They now must explain to voters impatient for action in Washington why nothing was their best choice.

In a hint of the coming Republican line of argument, Sens. Judd Gregg, R-N.H., and Jon Kyl, R-Ariz., said the new law would push the United States to a ‘‘European-style’’ government.

More than a dozen Republican senators introduced legislation to repeal the law that Sen. Jim DeMint, R-S.C., said would ‘‘force taxpayer funding of abortions, raise health costs, hike taxes, cut Medicare, raid Social Security and put bureaucrats between patients and their doctors.’’

‘‘Repeal and replace,’’ Senate Minority Leader Mitch McConnell, R-Ky., told reporters.

And attorneys general from 13 states acted on their opposition immediately, filing suit to stop the overhaul just minutes after the bill signing. Florida Attorney General Bill McCollum took the lead in the lawsuit that contends the legislation is unconstitutional, joined by colleagues from South Carolina, Nebraska, Texas, Michigan, Utah, Pennsylvania, Alabama, South Dakota, Louisiana, Idaho, Washington and Colorado. Other GOP attorneys general may join the lawsuit later or sue separately.

Obama made clear that the Republican offensive will not go unanswered. His larger, second event, held in a vast Interior Department auditorium, had a more combative feel. He accused Republicans of telling ‘‘lies.’’

‘‘Those fighting change are still out there, still making a lot of noise about what this reform means,’’ he said. ‘‘Look it up for yourself.’’

Starting with a Thursday trip to Iowa City, where as a presidential candidate he announced his health care plan in May 2007, Obama intends to emphasize the law’s most immediate impacts, including the ability of young adults to remain on their parents’ health plans and a ban on insurers denying coverage to sick children.

Even as the celebration proceeded in Washington, Congress labored to complete the overhaul with a companion measure containing changes demanded as a condition of House Democrats’ approval. The Senate was poised to consider that bill, with Democratic leaders hoping for its completion by week’s end.

‘‘We are going to make a good law signed by the president even better,’’ Majority Leader Harry Reid, D-Nev., declared on the Senate floor.