THE VILLAGES, Fla. — Who loves Medicare more? President Barack Obama and Mitt Romney’s running mate vied for that distinction Saturday as Medicare became the latest flashpoint in a presidential campaign of flying elbows.

The issue is dicey for both sides: Obama is steering billions from the entitlement to help pay for the expansion of coverage under his health care law; Paul Ryan is a champion of overhauling Medicare to make the traditional program no longer the mainstay for tomorrow’s seniors —  just one of many old-age health insurance choices.

But that didn’t stop them from going head on.

On a day Romney devoted to raising campaign cash in Massachusetts, Ryan accused Obama of raiding the Medicare “piggybank” to pay for his health care overhaul and he warned starkly that hospitals and nursing homes may close as a result. The Wisconsin congressman introduced his 78-year-old mother to an audience of seniors in Florida and passionately defended a program that has provided old-age security for two generations of his own family.

“She planned her retirement around this promise,” Ryan said as Betty Ryan Douglas looked on. “That’s a promise we have to keep.”

Campaigning in New Hampshire, Obama said it’s a promise that the Republican ticket would tear up.

“You would think they would avoid talking about Medicare, given the fact that both of them have proposed to voucherize the Medicare system,” he said in Windham. “But I guess they figure the best defense is to try to go on offense.

“So, New Hampshire, here is what you need to know: Since I have been in office, I have strengthened Medicare.”

He hammered the point again later in the day while campaigning in Rochester, N.H.

Said Ryan in Florida: “You want to know what Medicare is saying about this? From Medicare officials themselves: One out of 6 of our hospitals and our nursing homes will go out of business as a result of this,” meaning Obama’s Medicare cuts.

That was a far from exact reference to a 2010 analysis by Medicare chief actuary Richard Foster. He said then that roughly 15 percent of hospitals and nursing homes that provide Medicare services could “become unprofitable” over a decade — not necessarily go out of business — thanks to cuts in payments from the government under the health care law.

But Foster’s analysis also said the law would improve key Medicare benefits, solve the “doughnut hole” gap in coverage for seniors, expand health insurance to millions more people, reduce the federal budget deficit and extend the solvency of the government’s hospital insurance trust fund by up to 12 years. Hospitals remain largely on board with the health care law, without apparent fear of closing.

Ryan, a deficit hawk and the House Republicans’ chief budget writer, has stood out in Washington for laying out tough spending choices that many lawmakers in both parties avoid. So it was almost inevitable that his selection as running mate would vault Medicare to the top of the campaign debate.

Democrats say it’s a debate they are glad to have because voters tend to trust them more than Republicans on the big social entitlements. But Obama has vulnerabilities, too, given the Medicare cuts he pushed to expand health insurance for the nation and to keep the costs of doing so in line.

The Obama campaign recognizes that Romney and Ryan have been pre-emptive. The likely Republican ticket tried to neutralize the usual Democratic criticism on Medicare by striking first with a Medicare ad and with their criticism of Obama’s health law. “They are being dishonest about my plan because they can’t sell their plan,” the president said Saturday.

Ryan’s proposal in Congress would encourage future retirees to consider private coverage that the government would help pay for through a voucher-like system, while keeping the traditional program as an option.

According to the nonpartisan Congressional Budget Office, Medicare over time would spend thousands less per senior under the Ryan plan than under current policy. Critics say that would shift heavy costs to individual retirees. The government could always spend more than anticipated to meet changing realities, but at the cost of deeper deficits.

In New Hampshire, Obama cast the choice on Election Day as one between two fundamentally different approaches to the government’s responsibility to its citizens. His approach of portraying Romney’s tax and economic plans as a giveaway to the rich was familiar, but seemed to have a particularly sharp bite.

“They’ve been trying to sell this trickle-down snake oil before,” he told his audience in Windham. “It did not work then. It will not work now. It will not reduce the deficit, it will not create jobs. It’s the wrong direction for America.”

In Massachusetts, Romney told reporters on Martha’s Vineyard that he wishes he could spend more time campaigning in competitive states but must raise money at a furious pace because Obama broke all barriers four years ago.

“That’s the challenge with a president who blew through the federal spending limits,” he said. “Campaigns now have to spend a disproportionate amount of time fundraising. You appreciate all the help you get, but you wish you could spend more time on the campaign trail.”

Asked if campaigns ads are not already saturating the airwaves in swing states, Romney replied, “80-some-odd days to go.”

His staff estimated Romney will raise nearly $7 million from fundraising events held Friday and Saturday in Boston, Long Island and the Massachusetts resort areas of Martha’s Vineyard, Cape Cod and Nantucket.

Romney took a 10-minute break from fundraising Saturday to shake hands and pose for pictures at Millie’s Restaurant in Nantucket, Mass.

Romney bought ice cream for several staff members, shook hands with adults and stroked a baby’s head.

Alec Gavenda, 13, of Summit, N.J., marched up to Romney and introduced himself and his family to the candidate, who asked several questions about the vacationing group.

Greg Gavenda, 12, told Romney, “my brother has Down Syndrome,” to which Romney smiled and said softly, “I figured that.”

The boys’ father, T.J. Gavenda, told the former governor, “We just ordered our Romney-Ryan yard sign.”

But as the campaign entourage left the restaurant, a less-friendly man shouted demands that Romney release five years of personal tax returns.

Although Romney was governor of Massachusetts, he’s conceded that Obama’s almost certain to win the state in November.

Ryan, too, took a break — to raise money. At an evening reception on Florida’s Treasure Island, Ryan drew a crowd of 200 people and raised another $1 million.

Speaking to donors who paid as much as $50,000 to have dinner with him, Ryan compared the United States with Europe, where a financial crisis has led to cuts in benefits for retirees. He said lawmakers there delayed action even though they saw impending problems, and seniors paid the price.

“They ran out of road to kick the can down, and now they have a debt crisis,” Ryan said.

Ryan warned the same could happen here if the country doesn’t get its hands around its own affairs.

Ryan’s stop Saturday at the gated retirement cluster known as The Villages was familiar ground for presidential candidates. Florida has the highest concentration of voters over 65 in the country, with some 17 percent of Floridians fall into that group. Betty Ryan Douglas spends part of her year in Broward County’s Lauderdale-by-the-Sea community and has been registered to vote in Florida since 1997.