President Barack Obama and Mitt Romney set their sights Wednesday on an epic clash for the White House after the Republican tightened his grip on his party race with a crushing Florida primary win.

Romney's victory over Newt Gingrich restored him as the hot favorite to head the Republican ticket in November's election, and revived the prospect of the showdown with the Democratic president that the Obama camp has always expected.

The former Massachusetts governor trained his fire not on his Republican rivals but directly on Obama in his victory speech Tuesday night, driving home the narrative that only he could restore America's historic mission.

"Mr President, you were elected to lead. You chose to follow, and now it's time for you to get out of the way," a jubilant Romney said in Tampa.

"President Obama wants to fundamentally transform America and make it something perhaps we wouldn't recognize. I want to restore to America the values and principles that made us the hope of the Earth."

The Obama campaign quickly hit back on Wednesday, arguing that a toxic Republican nominating melee was weakening Romney's general election prospects.

Senior strategist Stephanie Cutter argued that Romney's Florida victory had come at a "steep price" -- not just in terms of the $15 million he and allies reportedly splashed on a fearsome negative advertising blast at Gingrich.

"Second, and more ominously for Romney, his unprecedentedly negative, far-right Florida campaign continued to damage him among the swing voters he would need in November," Cutter wrote in a campaign memo.

Romney had demonstrated a "unique ability to push key voting constituencies away from his candidacy rather than attract them, Cutter wrote, as Democrats root for a long and damaging Republican nominating contest.

Recent polls of vital swing voters do support the view that Romney would benefit by ending the bitter Republican race as soon as possible.

An ABC News/Washington Post survey last week showed that Romney's unfavorable rating among independent voters climbed to 51 percent while his favorable rating fell to 23 percent.

Obama, though, has his own problems with independents, a majority of whom backed his 2008 presidential campaign: a New York Times/CBS poll last month put his approval rating with the key bloc at 31 percent.

Romney's victory in Florida demonstrated his ability to run a well drilled and lucratively funded statewide campaign and shrewd targeting of key territory in central Florida which is home to most of the state's swing voters.

Florida has 29 of the 270 electoral votes needed to win the presidency and, given how the political map is shaking out, is a must-win for Romney if he is to capture the White House.

Obama has uncertain prospects in the state, given an unemployment rate of nearly 10 percent and a tsunami of mortgage foreclosures which have hammered the savings and security of the middle class.

The president could lose Florida and still win the presidency, according to most projections, but his margin of victory would be considerably narrowed, leaving cliffhanger races in other swing states even more crucial.

A Quinnipiac University poll last week showed Obama and Romney locked in tie in the state. Recent national general election polls show Obama with a lead of only a few points over the Republican.

Romney on Wednesday, seeking to project an air of inevitability to effectively nudge his main rival Gingrich out of the Republican race, targeted Obama's economic record.

"The truth is he has failed the American people. He's lost more jobs during his presidency than any other president, I believe in history," Romney told CNN.

"Home values (are) down during his presidency. He doesn't want to talk about that, but that's what we're going to talk about."

Obama knows that barring a surprise explosion of growth, he will probably lose November's election if it becomes a referendum on his economic record, so he is trying to frame an alternative narrative for voters.

Contrary to Romney's claim, Obama did talk about rock-bottom house prices on Wednesday.

He appeared in suburban Virginia, another swing state, touting a new plan to allow responsible homeowners to refinance underwater mortgages, part of a push for a new economy that benefits everyone, not just the rich.

"I've been saying this is a make-or-break moment for the middle class. And this housing crisis struck right at the heart of what it means to be middle class in America: our homes," Obama said.