President Obama opened a two-day tour of Florida on Thursday with a direct appeal to the state’s sizable constituency of seniors, as polls show the presidential race tightening in this crucial battleground.

At his first stop here at the Prime Osborn Convention Center, Obama accused Republican rival Mitt Romney of wanting to dismantle Medicare, hoping to cast doubt about Romney in the minds of Florida’s 3.3 million seniors, who make up 30 percent of the state’s electorate. After being introduced by a 70-year-old Navy veteran, Obama told a crowd of 3,000 that Romney supports turning Medicare into a privatized voucher program that could raise medical costs on the elderly.

“That’s the wrong way to go,” Obama said. “It’s wrong to ask seniors to pay more in Medicare just so millionaires and billionaires can pay less in taxes. We shouldn’t be squeezing more money from seniors who are just barely getting by right now.”

Romney’s response in the past to similar attacks has been ­to ­accuse Obama of stripping $500 billion from Medicare to pay for his health-care legislation. This is a reference to the difference, over 10 years, between anticipated Medicare spending and changes made by the law to reduce spending.

Florida remains the biggest swing-state prize in this election, with 29 electoral votes, and is considered a must-win for Romney, who is expected to accept the GOP nomination at the Republican National Convention in Tampa in August. A recent Miami Herald-Tampa Bay Times poll found that 46 percent of the state’s likely voters backed Obama and 45 percent favored Romney, with 7 percent still undecided, almost erasing an advantage of four to 10 points that the president had held for months.

Obama’s popularity has been buffeted by Florida’s 8.6 unemployment rate, which is above the national average, and the housing market’s downturn, which hit portions of the state particularly hard.

The same newspaper poll also showed that 52 percent of Florida voters view Obama’s health-care act unfavorably, compared with 43 percent who support it. In his remarks in Jacksonville, Obama touted the legislation, saying it would help seniors reduce their prescription drug costs.

“The Supreme Court has spoken,” Obama said, referring to the court’s 5 to 4 decision to uphold the individual mandate.

In Boston, Romney stopped by his campaign headquarters to address his staff members, thanking them for their hard work.

A few hours later in Roxbury, Mass., Romney resumed his attacks on Obama at an unscheduled campaign stop, painting him as out of touch on the economy and hostile to small businesses.

Speaking at Middlesex Truck and Coach, a vehicle-repair facility, Romney for the second straight day pointed out that Obama’s jobs council has not held a meeting in six months.

Picking up on recent Republican attacks on the president, Romney reminded his audience of an Obama rally last week in Virginia during which the president used the phrase “you didn’t build that” while arguing that the government helps the private sector by building the roads, bridges and education system that businesses rely on.

“Somebody helped to create this unbelievable American system that we have that allowed you to thrive,” Obama said Friday. “If you’ve got a business, you didn’t build that.”

In television ads, Republicans have seized on the phrase to make the case that Obama prefers big government to free-market capitalism and entrepreneurship.

“It wasn’t a gaffe. It was, instead, his ideology,” Romney said.

In a conference call with reporters this week, Obama campaign spokesman Ben LaBolt said Romney’s attack on Obama’s Virginia remarks is an attempt “to change the subject from questions coming at his campaign about whether or not he is responsible for outsourcing American jobs or bankrupting companies. He’s ripping the president’s remarks out of context.”

Romney has aimed to use the offensive to bounce back from more than a week of pressure from Democrats, who have raised questions about Romney’s tenure at Bain Capital and about his refusal to release tax records predating 2010.

While national polls continue to show Obama and Romney running neck and neck, a new USA Today-Gallup poll showed that Democrats’ focus on Romney’s tax returns appears to have resonated with the public.

According to the survey, 54 percent of adults nationwide think that Romney should release more than two years of tax returns. The 54 percent figure includes 75 percent of Democrats, 53 percent of independents and 30 percent of Republicans.

During his two days in Florida, Obama planned to stop in Palm Beach on Thursday before traveling to Fort Myers and Orlando on Friday. The areas represent key territory for Obama, who carried Florida in 2008 but lost to John McCain in Jacksonville, as well as Lee and Collier counties, in the region where Fort Myers is located.

Obama was last in Fort Myers in February 2009, shortly after taking office, to announce the American Recovery Act, a trip punctuated by his infamous embrace of then-Gov. Charlie Crist (R). That moment was later used by Republican Marco Rubio to mock Crist’s support of the Obama administration stimulus in his successful campaign for a Senate seat.

Rubio was not impressed by Obama’s jaunt through his state Thursday, observing on his Twitter account that the president was waging class warfare during his Jacksonville speech.

“Parts of it sound like speech by left-wing 3rd world leader,” Rubio wrote in a tweet.

Sonmez reported from Boston.