LACONIA, N.H. — His credibility under attack, Republican presidential hopeful Mitt Romney insisted on Friday that he had “no role whatsoever in the management” of a private equity firm after early 1999, and demanded that President Barack Obama apologize for campaign aides who persist in alleging otherwise.

“This is simply beneath the dignity of the presidency of the United States,” Romney said in an interview on ABC, one of several he granted to network and cable stations in hopes of extinguishing the controversy.

Under pressure from Democrats and even some Republicans to release tax returns going back several years, Romney indicated he wouldn’t do so.

“You can never satisfy the opposition research team of the Obama organization,” he told CBS.

Romney said after he left Bain Capital he retained ownership “until we were able to negotiate a departure” from the company he had founded. “I had no role whatsoever in the management of Bain Capital after February of 1999,” he said, adding that officials at the company and independent fact-checkers had said the same thing.

He also said, “I was an owner, and being a shareholder doesn’t mean you’re running the business.” He said he couldn’t recall attending any Bain management meetings after he moved to Salt Lake City to oversee the Olympic Games.

The precise role Romney played at the firm between 1999 and 2001 is important not only because critics have raised questions about his truthfulness, but also because Bain was sending jobs overseas during that period.

That, in turn, goes to the core issue of the race for the White House in dreary economic times, Romney’s claim that as a former businessman, he has the ability to create jobs and finally pull the country out of a downturn that has lingered throughout Obama’s term. The Obama campaign has criticized Romney as running a firm that pioneered job outsourcing.

Some Securities and Exchange Commission documents have surfaced suggesting Romney played an active role in the Boston-based company through 2002. The filings with the SEC place Romney in charge of Bain Capital from 1999 to 2001, the period in which it outsourced jobs and ran companies that fell into bankruptcy.

Obama himself had stepped into the controversy a few hours before Romney’s comments were aired.

“Now, my understanding is that Mr. Romney attested to the SEC multiple times that he was the chairman, CEO and president of Bain Capital. And I think most Americans figure if you’re the chairman, CEO and president of a company that you are responsible for what the company does,” Obama said in an interview with WJLA-TV in Virginia as he campaigned across the battleground state.

The president said the questions that have been raised in numerous media reports and highlighted by his own campaign aides were a legitimate part of the race for the White House.

“Ultimately, I think, Mr. Romney is going to have to answer those questions because if he aspires to being president, one of the things you learn is you’re ultimately responsible for the conduct of your operations,” the president said.

But Romney said that was “Chicago-style politics at its worst,” and accused the president and his campaign of trying to shift attention from the persistently sluggish economy and unemployment of 8 percent or higher for more than 40 months.

Romney was particularly harsh when asked about a claim by Obama deputy campaign manager Stephanie Cutter suggesting he might be guilty of a felony for misrepresenting his position at Bain to the SEC.

“Is this the level that the Obama campaign is willing to stoop to?” he responded on CNN. “Is this up to the standards expected of the presidency of the United States?”

The Obama campaign said it would not apologize.

Romney acknowledged that he would have benefitted financially from Bain’s operations even after he left management of the firm to others. That could open him up to criticism that he gained from investment in companies that sent jobs overseas.

“All of the investors participate in the success or failure of various investments, just like you do as a shareholder of an enterprise,” Romney said.

One Romney aide said earlier in the week that any suggestion that Romney had shipped jobs overseas was a lie, and the campaign has said repeatedly the break with the private equity firm came in 1999.

Yet documents surfaced for the second straight day that seemed to indicate Romney played an active management role in Bain Capital after that date, when he and company officials say he left the firm to run the Olympic.

Annual reports filed by Bain Capital with the state of Massachusetts said he had a role in the firm as late as 2001. The company’s 2002 report did not list Romney among the firm’s managing partners, suggesting he was no longer with Bain by that point.

Obama spent much of his day challenging Romney over taxes and spending, telling one audience that if Republicans are unwilling to let tax cuts lapse for the wealthiest Americans, they’re “not serious” about reducing the deficit.

Appearing in Hampton Roads, Va., Obama renewed calls to extend Bush-era tax breaks for those earning $250,000 or less while the two sides argue about higher earners, affecting the top 2 percent of Americans. But he charged Republicans with balking, and holding middle-class cuts “hostage.”

“If you say you want to bring down the deficit, but you’re not willing to let tax cuts lapse for the top 2 percent, it tells me you’re not serious about deficit reduction,” Obama said at a campaign rally. He said lawmakers should “go ahead and help middle-class families right now. And so far, I have not gotten an OK from the other side on that. And that tells me I guess they’re not that serious about deficit reduction.”

Romney and other Republicans argue that raising taxes on anyone would be a mistake given the fragile state of the recovery.

But for now, the dispute over outsourcing and the timing of Romney’s departure from the firm he created took a central role in the race.

Separately, Bain Capital issued a statement saying that Romney “remained the sole stockholder for a time while formal ownership was being documented and transferred to the group of partners who took over management of the firm in 1999.”

The dispute expanded to the television ad wars already under way.

Romney unveiled a scathing commercial this week declaring there was no truth to the accusations and saying that in 2008, “candidate Obama lied about Hillary Clinton.”  It showed the secretary of state — then Obama’s rival for the Democratic nomination — saying “shame on you, Barack Obama.”

Obama responded on Friday with an ad that accused the Republican of favoring a 25 percent tax cut for millionaires, tax breaks for oil companies and corporations that move jobs overseas and a tax increase for working families. By contrast, it says, the president wants “the wealthy to pay a little more so the middle class pays less.”