The campaign debate over President Obama's "you didn't build that" remark raged on Wednesday even as Mitt Romney arrived in London for his overseas trip -- with Obama conducting damage control in a new TV ad and Romney's campaign hammering back with, "he said it, he meant it."

The president over the past several days has argued that Republicans twisted his words and took them out of context. While Romney claims Obama was talking about businesses when he said, "if you've got a business, you didn't build that," Obama now says he was actually trying to make the point that business owners "didn't build" roads and bridges -- something he had referred to earlier in the speech.

The president was out with a new TV ad making that case, and refuting recent GOP ads. The ad shows Obama looking straight into the camera, saying: "Those ads, taking my words about small business out of context, they're flat-out wrong. Of course Americans build their own businesses."

But the Romney campaign doubled down on its criticism, accusing Obama of trying to walk back what he said.

"He said it, he meant it," the campaign said in a statement Wednesday morning.

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"When President Obama declared that Americans didn't build their businesses, he wasn't saying anything extraordinary - he was just saying what he actually believed," spokeswoman Amanda Henneberg said. "After three and a half years of liberal policies that have grown the size of government, President Obama's record is clear."

The Romney campaign also announced it would be holding two-dozen "We Did Build This" events in battleground states on Wednesday, featuring local business owners.

The Romney campaign is keeping the controversy alive, as the GOP candidate himself arrives in London for a nearly weeklong tour of Britain, Poland and Israel.

The "you didn't build that" quote has provided steady fodder for Romney and Republican campaign groups, who have used the snippet to suggest Obama doesn't give credit to America's entrepreneurs.

That debate, though, has drifted into an argument over the meaning of "that." Was Obama talking about businesses, or roads and bridges?

Here's what Obama said earlier this month.

"If you were successful, somebody along the line gave you some help. There was a great teacher somewhere in your life. Somebody helped to create this unbelievable American 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. The Internet didn't get invented on its own. Government research created the Internet so that all the companies could make money off the Internet."

Republicans have nevertheless taken issue with the context of Obama's comment, not just the "you didn't build that" line.

"The context is worse than the quote," Romney told CNBC.

The Republican National Committee released a web video Wednesday highlighting the rest of what Obama said. The video was titled, "The More Context You Get, The Worse It Sounds."