With his forceful denial of charges that he would raise taxes on the middle class, Mitt Romney used Wednesday’s debate to launch an aggressive new effort to regain his footing in the battle over taxes.

In one of the debate’s first exchanges, the Republican presidential nominee directly challenged President Obama’s assertion that Romney’s tax plan would finance big new breaks for the wealthy by wiping out popular deductions for those who earn less than $250,000 a year.

“I know that you and your running mate keep saying that. I know it’s a popular thing to say with a lot of people, but it’s not the case,” Romney said onstage in Denver. “I will not reduce the taxes paid by high-income Americans. And . . . I will not, under any circumstances, raise taxes on middle-income families. I will lower taxes on middle-income families.”

Hours later, Romney released a new TV ad arguing that Obama, not Romney, is the bigger threat to middle-class pocketbooks — the second this week. Neither spot offers new information about how Romney would pay for his tax plan, which is heavy on promises but light on details. Instead, the ads seek to shift the conversation to more comfortable territory for Republicans: Which candidate is more likely to raise taxes?

Vice President Biden appeared to play into Romney’s hands on that front Thursday, acknowledging that he and Obama want to let taxes rise by $1 trillion for the nation’s wealthiest households over the next decade.

A study looks at the effects of Mitt Romney's tax proposals.

“Yes, we do,” Biden said at an event in Iowa. ‘‘On top of the trillions of dollars in spending that we have already cut, we’re going to ask the wealthy to pay more. My heart breaks. Come on, man. . . . That’s not a tax raise. That’s called fairness where I come from.”

The pushback from Romney comes as Republicans are widely concerned that he is doing a poor job of defending his tax plan, the centerpiece of his agenda for sparking economic growth and creating jobs. Public opinion polls have shown that Obama has held a consistent advantage on the question of which candidate voters trust more to handle taxes. A Washington Post poll last month in the critical swing state of Ohio found Obama ahead on the tax question by 17 percentage points.

Asked about the polls on Thursday, Romney senior adviser Ed Gillespie said, “The Obama campaign has spent a ton of money trying to convince voters that Governor Romney would raise taxes on the middle class.”

Gillespie said Romney is fighting back now — fully two months after Obama first launched his attack — to take advantage of the national exposure of the debate.

“We want to weigh in heavily with ads on the subject after 60 million people had a chance to hear him set the record straight,” Gillespie said.

Romney’s sharp performance cheered conservatives, who worried that Obama’s edge on the tax issue could do lasting damage to the cause of tax revision, which has been gaining momentum in both parties.

“He made the point that this is a pro-growth tax cut,” said Grover Norquist, president of Americans for Tax Reform. “I would have preferred this to be the conversation for the last six months. But I’ll take that it’s the issue for the last month of the campaign.”

Romney’s plan calls for cutting income tax rates by 20 percent for people at all income levels, repealing the estate tax and the alternative minimum tax, and wiping out capital gains taxes for middle-class families.

Those cuts would drain nearly $5 trillion out of the Treasury over the next decade, according to budget analysts. Romney has vowed to replace the lost revenue by scaling back “loopholes and deductions,” but he has declined to say which ones.

That lack of detail created a political opening for Obama, who in August latched on to a study by the nonpartisan Tax Policy Center that declared Romney’s goals for tax revisions “not mathematically possible.”

The Tax Policy Center, a joint project of the Urban Institute and the Brookings Institution, found that Romney’s rate cuts would reduce tax collections from individuals by $360 billion in 2015. To avoid increasing deficits, it said, the plan would have to generate an equal sum by cutting tax breaks for mortgage interest, employer-provided health care, medical expenses and child care — all breaks that benefit the middle class.

Even if Americans in the upper tax brackets were targeted first, the study found, “there would still be a shift in the tax burden of roughly $86 billion [a year] from those making over $200,000 to those making less” than that. As a result, millionaires would get an $87,000 tax cut, on average, while 95 percent of the population’s taxes would rise about 1.2 percent, an average of $500 a year.

In television ads and on the campaign trail, Obama has cited the study relentlessly, with little response from Romney until this week. “Amid this barrage, Mr. Romney has . . . played dumb, as in silent,” the Wall Street Journal editorial page said on the morning of the debate.

Worried academics and think-tank fellows came to Romney’s defense, arguing that Romney could pay for his tax cuts in any number of ways without harming the middle class. He could end the exemption for interest earned on state and local bonds, for example, or tax the buildup of value inside life insurance policies.

In the meantime, the Romney campaign is citing a study from the conservative American Enterprise Institute to accuse Obama of a secret plot to tax the middle class more. The ad unveiled on Thursday claims that, in order to make payments on the national debt over the next decade, “Barack Obama and the liberals will raise taxes on the middle class by $4,000.”

Obama campaign spokeswoman Lis Smith said in a statement that the ad “gets an F in honesty” and “has nothing to do with the president’s proposals.”

Romney’s newly aggressive stance appears to be helping his cause, at least initially. A CBS News instant survey of uncommitted voters found that they favored Obama by a significant margin on the tax issue going into Wednesday’s debate. Immediately afterward, the numbers flipped.