WASHINGTON -- President Barack Obama has signed into law a sweeping overhaul of U.S. health care in a defining moment of his presidency, but one last chapter in the epic struggle is still playing out in the Senate.

Senators are debating a package of fixes to the new health reform law, demanded by House Democrats as their price for passing the nearly $1 trillion overhaul legislation that will extend coverage to 32 million uninsured Americans over the next decade.

Obama signed the bill on Tuesday, declaring "a new season in America" as he sealed a victory denied to a line of presidents stretching back nearly half a century. Failure would have weakened him and endangered other issues on the president's ambitious domestic agenda, including immigration reform and climate change legislation.

The fix-it bill under consideration in the Senate eliminates special deals for some individual states from the new law, softens a tax on high-cost insurance plans that was repugnant to organized labor, provides more expansive subsidies to lower-income people to purchase insurance, and offers more generous prescription drug coverage to seniors, among other changes.

Its approval at the end of this week is virtually assured, since it's being debated under fast-track budget rules that allow passage with a simple majority instead of the 60 votes usually required for action in the 100-seat Senate. Democrats control 59 Senate seats.

That didn't stop Republicans, who are unanimously opposed, from using the floor debate that began Tuesday afternoon in the Senate as an opportunity to repeat the accusations they have lobbed at Obama's health legislation for the past year: that it raises taxes, slashes Medicare coverage for seniors, and includes a burdensome and constitutionally questionable requirement for nearly all Americans to carry health insurance.

Republicans came up with some new arguments too, including an amendment offered by Sen. Tom Coburn that would prohibit sex offenders from getting Viagra prescriptions under federal health programs.

Democratic Sen. Dick Durbin dismissed that as a "gotcha amendment" designed to be difficult for Democrats to oppose.

The main suspense surrounding this week's debate is whether the fix-it bill can emerge from the Senate unchanged. If it does, it can go straight to the president for his signature, since it's already passed the House of Representatives. If the Senate changes it even in a minor way, the legislation would have to go back to the House to be passed again, a prospect House leaders are prepared for but say they don't expect.

If there are only minor changes the House would be almost certain to pass the bill again with little trouble, but if Republicans succeed in knocking out a significant provision or attaching a substantive amendment there could be difficulties in the House, where the legislation passed very narrowly Sunday night. Democratic leaders in the House and Senate say they have scrubbed the fix-it bill thoroughly to ensure that will not happen.

Republicans are introducing an array of politically sticky amendments such as Coburn's and another that would stipulate that Obama himself must get health coverage through a new purchasing exchange to be established under the health law.

The Republicans also are planning to raise points of order under rules requiring that provisions of the fix-it bill must have a budgetary impact. If Republicans argue that something doesn't and the Senate parliamentarian rules in their favor, the provision in question likely would be knocked out.

For Republicans, making it more difficult for Democrats to pass the fix-it bill is about the end of the road for congressional roadblocks against Obama's yearlong overhaul drive that will impact one-sixth of the U.S. economy. But opponents already have launched a campaign from the outside, with 13 state attorneys general -- all but one of them Republicans -- suing Tuesday to overturn the legislation on grounds it is unconstitutional to force people to get health coverage.

And Senate Minority Leader Mitch McConnell served notice Tuesday of the Republicans' continued campaign against the legislation leading up to the November election when control of Congress will be at stake. "The slogan will be 'repeal and replace,' 'repeal and replace,"' McConnell said.

Opinion polls show the American public remains skeptical. Obama is planning a number of appearances to promote the plan, starting with a trip to the politically pivotal Midwest state of Iowa on Thursday. He intends to emphasize the law's most immediate impacts, including the ability of young adults up to age 26 to remain on their parents' health plans.

Obama planned to sign an executive order Wednesday affirming existing law against federal funding of abortions, except in cases of rape, incest or danger to the woman's life. A critical bloc of anti-abortion Democrats in the House had pledged to vote against the health care package unless given greater assurances that it would not amend current law.

In a last-minute deal, Obama agreed to issue the order to get their votes for passage of the legislation.