Republican presidential hopeful Mitt Romney launched a concerted push Wednesday to depict President Obama as hostile to small business, using the infamous snippet from one of Obama’s speeches now known in political shorthand as “You Didn’t Build That.”

The president’s campaign countered with a new television ad saying that his remarks were flagrantly wrenched out of context. The dispute is the latest example of a campaign plucking lines from an opponent’s speech in an effort to support and reinforce a broader, negative narrative that it hopes will resonate with voters. In this case, Republicans contend that the remarks capture Obama’s identity as a leader who neither understands nor respects business and instinctively looks to government for solutions.

The comments came from the tail end of 40-minute campaign speech at a Roanoke fire station on July 13.

In that speech,Obama repeated his opposition to Romney’s proposal to cut taxes for the wealthiest Americans, saying there are successful people who agree and who want instead to forgo tax breaks and “give something back.” He added that those who succeed in business are aided by personal mentors and government policies that support infrastructure and technology.

“If you were successful, somebody along the line gave you some help,” Obama said. “There was a great teacher somewhere in your life. Somebody helped to create this unbelievable system that we have that allowed you to thrive. Somebody invested in roads and bridges. If you’ve got a business, you didn’t build that. Somebody else made that happen.”

Obama was referring to the roads and bridges — not businesses — as having been built by government.

The theme is not a new one. Massachusetts Democratic Senate candidate Elizabeth Warren triggered a similar controversy last year when she said: “There is nobody in this country who got rich on his own — nobody. You built a factory out there? Good for you. But I want to be clear. You moved your goods to market on the roads the rest of us paid for. You hired workers the rest of us paid to educate.”

But coming from a sitting president, the comments were of another magnitude. While taking liberties with the context, the Romney campaign seized on “You Didn’t Build That” as a rallying cry Wednesday. It staged two-dozen “We Did Build This” rallies, including one in Richmond, where small-business owners joined Gov. Robert McDonnell in assailing the president.

“President Obama is wrong,’’ said Melissa Ball of Ball Office Products. “Americans do build their own business and we need a president who believes that as well.” Ball said she wanted to invite Obama “to hear about the struggles of real-life business owners.” According to the federal database Web site, Ball has received several small federal contracts in the past, and Democrats used information like that to push back hard.

In his new ad Wednesday, Obama, looking directly into the camera, said such Republican assertions were “flat-out wrong.”

“Of course Americans build their own businesses. Every day, hardworking people sacrifice to meet a payroll, create jobs and make our economy run,” he said. “And what I said was that we need to stand behind them, as America always has, by investing in education and training, roads and bridges, research and technology.”

Romney has also had his rhetoric yanked out of context. Campaigning in New Hampshire in January, Romney told the Nashua Chamber of Commerce, “I like being able to fire people who provide services to me.” It was part of a longer answer to a question from the audience about health-care policy — specifically, being able to change insurance companies. Just prior to the “fire” comment, Romney said: “I want individuals to have their own insurance. That means the insurance company will have an incentive to keep you healthy. It also means that if you don’t like what they do, you could fire them.”

But, in isolation, ricocheting around the Web, the quote helped opponents reinforce an image of Romney as a remorseless, corporate predator.

The Obama campaign pointed to a series of initiatives that the administration has pursued to help small business, including tax cuts, improved access to capital and an overhaul of the patent system that officials said accelerates processing.

Small-business leaders say Obama’s record is mixed. Todd McCracken, president of the National Small Business Association, said that for some small firms, the administration has been a boon. For businesses that export, for example, McCracken said that reauthorization of the Export-Import Bank Act will help ensure a level playing field. He also described his 65,000 member companies as being “quite comfortable” with the individual insurance mandate at the heart of the Affordable Care Act. The one-year extension of the George W. Bush-era tax cuts for incomes under $250,000 is also a plus, he said.

But McCracken said Obama has not been aggressive enough in compelling banks to loosen lending policies. And in many instances, he said, tax relief is secondary to the regulatory burdens that leave many small businesses uncertain.

In Old Town Manassas, several small-business owners said Obama’s comments — if they had heard about them — had not elicited a strong reaction one way or the other.

“I work hard for what I’ve got, but I’ve had people along the way help me,” said Matt Brower, owner of Simply Sweet on Main, a new ice-cream and coffee shop.

Brower, 32, said that he generally tries to stay out of politics but that the president’s comments about small businesses “didn’t turn me the wrong way.” He credited the city government for allowing him to sell coffee in the nearby train station during colder months. While he had to sign a contract with the city, officials did not charge him anything to set up or operate his stand — a big help since he just started the business two years ago, he said.

When Charles Gilliam opened his restaurant in 1998 in Old Town, he was “happy if someone would come in every day.” Now, Okra’s, which serves Cajun and Creole fare, is thriving, and Gilliam is looking at opening a second restaurant elsewhere.

A picture of first lady Michelle Obama is in the foyer, and Gilliam proudly points to the spot where she and her children came in for a bite one day a couple of years ago.

“It was exciting to know the highest office in the country knows about what we do,” Gilliam said.

In his business, Gilliam said, he “hasn’t asked for help from government.” If help was offered, “we wouldn’t take it,” he said.

In recent years, city officials have made the sidewalks broader on Battle Street, where Okra’s sits caddy-corner. That has allowed restaurants to put in outside patio seating and offer live music.

“We’d still be doing very well” without the local-government-backed sidewalk project, he said. “But we’re grateful to have it.”

Anita Kumar in Richmond and Jeremy Borden in Manassas contributed to this report.