Republican presidential candidate Mitt Romney, left, and running mate Paul Ryan at a campaign rally at the Harris Pavilion in Manassas, Va., on Aug. 11 (SAUL LOEB/AFP/GETTY IMAGES)

Mitt Romney’s presidential campaign Friday rejected a new call from the Obama campaign to release five years of tax returns, while trumpeting a surge in support for the Republican ticket since Romney chose Rep. Paul Ryan (R-Wis.) as his running mate.

The sparring over Romney’s tax returns came as President Obama defended his record on Medicare in a new campaign advertisement that accused Romney and Ryan of undermining the program, which provides health insurance to nearly 50 million elderly Americans and younger people with disabilities.

In a letter Friday to Romney campaign manager Matt Rhoades, Obama reelection campaign manager Jim Messina pledged that if Romney released five years of tax returns filed from 2007 to 2012, the Obama campaign in turn would stop criticizing the Republican candidate for not making public more of his returns. Messina said the offer addresses Romney’s apparent “fears that the more he offers, the more our campaign will demand that he provide.”

Messina said the request “is surely not unreasonable,” noting that the five-year period would span the years that Romney has been a presidential candidate and noting that other White House hopefuls, including Romney’s father, have released more returns in the past.

Messina said Romney “would have to release only three more sets of returns in addition to the 2010 return he has released and the 2011 return he has pledged to provide.” Doing so “would also help answer outstanding questions raised by the one return he has released to date, such as the range in the effective rates paid, the foreign accounts maintained, the foreign investments made, and the types of tax shelters used,” he wrote.

Rhoades promptly rejected the offer, telling Messina in an e-mail: “It is clear that President Obama wants nothing more than to talk about Governor Romney’s tax returns instead of the issues that matter to voters, like putting Americans back to work, fixing the economy and reining in spending.”

He said that if Romney’s tax returns “are the core message of your campaign,” Obama will have ample time to discuss them between now and Election Day. “In the meantime, Governor Romney will continue to lay out his plans for a stronger middle class, to save Medicare, to put work back into welfare, and help the 23 million Americans struggling to find work in the Obama economy.”

The rival campaigns distributed the letter and the e-mail reply to reporters.

The Obama campaign’s challenge to Romney followed the former Massachusetts governor’s comment to reporters Thursday in South Carolina that he had paid a federal income tax rate of at least 13 percent in each of the past 10 years.

Romney focused Friday on raising money for his campaign. After an event Thursday in Greenville, S.C., that brought in $1.7 million, he held other fundraisers that night in Boston and scheduled a fundraiser Friday at the Sebonack Golf Club in Southampton, N.Y., the Associated Press reported.

Ryan was holding two campaign rallies Friday in Virginia: at Deep Run High School in Glen Allen with Gov. Robert F. McDonnell (R) and at West Springfield High School in Springfield. In the evening, the Wisconsin congressman is scheduled to attend fundraising events in Arlington, including a $15,000-a-person dinner.

In the week since Romney tapped Ryan to serve as his running mate, the GOP presidential ticket has seen a boost in the polls as well as in fundraising, social media and other metrics, Rhoades argued in a new campaign memo released Friday morning.

Since last Saturday’s announcement event in Norfolk, the Romney-Ryan campaign has raised more than $10 million online, Rhoades wrote. As much as $7.4 million of that amount was raised online in the first three days since Romney made his vice presidential pick public.

The campaign has raked in more than 124,800 donations, with an average donation of $81, and new donors account for 68 percent of the total amount raised since last Saturday, according to Rhoades.

The Romney campaign’s Web site has seen 2 million hits over the past week, 440,000 of which came from users accessing the site through mobile devices, Rhoades wrote. Romney now has 510,000 more Facebook fans — bringing his total to 4,360,000 — and has added 54,000 to his Twitter following, which now stands at about 861,000.

Ryan received an even greater social media bump — in large part, no doubt, because he was not as well known as Romney was before the pick. He has 860,000 more Facebook fans and 118,500 more Twitter followers than he did before being tapped for the No. 2 spot on the GOP ticket, according to Rhoades.

In terms of more traditional metrics, the campaign has signed up 45,000 new volunteers over the past week. Rhoades also pointed to some recent polls showing the national race in a dead heat and Romney gaining traction in the swing states of Ohio and Virginia.

Democrats have contended since the moment Romney’s move was announced that Ryan would prove to be more of a liability than an asset for the Republican presidential candidate. They pointed to Ryan’s budget blueprint and his plan to overhaul Medicare as just two examples of how they aim to put Republicans on the defensive.

Rhoades argued in his memo, however, that Romney’s choice of Ryan has “reshaped the race in a positive way” and that the GOP is very much on offense.

“The Obama team’s increasing vitriol is a sure sign that they’re rattled by the pick,” Rhoades wrote. “Unable to engage in an elevated policy debate, they’ve spent the past few days drowning in their own venom.”

A point underscoring the Romney camp’s argument that it can now go on the offensive against Obama is its scheduling of Ryan to deliver an address Saturday morning at a sprawling Florida retirement community, where his mother will be in the audience.

Apparently banking that the GOP ticket is vulnerable on Medicare, however, the Obama campaign in its new ad on the subject cited criticism of Ryan’s Medicare overhaul plan by the nonpartisan AARP organization, which represents retired people. The ad noted that the AARP had told lawmakers earlier this year that Ryan’s plan to turn Medicare into a voucher-like system would impose higher costs on seniors, AP reported.

The ad is running in eight states: New Hampshire, Virginia, North Carolina, Florida, Ohio, Iowa, Colorado and Nevada.