During a visit to Lansing, Mich., on Tuesday, Romney said the idea of a managed bankruptcy for GM and Chrysler had been his idea at the height of the economic crisis in 2009. "So I'll take a lot of credit for the fact that this industry's come back," he said.
Obama has argued that the managed bankruptcy could not have been possible without his decision to authorize a multi-billion dollar infusion of taxpayer cash to keep the companies afloat. Romney opposed federal government aid.
"I don't think anybody takes that seriously," he told Roberts of Romney's claim. "People remember his position, which was, 'Let's let Detroit go bankrupt' and his opposition to government involvement in making sure that GM and Chrysler didn't go under."
"And I - every businessperson and economist out there understands that at the time I had to make the decision, there was no private sector option. Nobody was opening up their wallets to lend money to GM and Chrysler," he said.
While a few conservative economists have rejected the notion that government funds were required, the consensus of leading economists is that situation was so dire that the companies could not alone acquire necessary funds to proceed through the process.
"The companies would have shut down and the bondholders would have been wiped out," said Mark Zandi, chief economist at Moody's Analytics. "Nearly all analysts at the time felt at the time that without government bailout - GM and Chrysler would have been liquidated."
Obama, who has made the revival of the auto industry a cornerstone of his re-election campaign, said a President Romney would have allowed the companies to succumb.
"We would have lost probably a million jobs throughout the Midwest," he told Roberts. "So the people who are in the Midwest - you know, you go take a poll of folks in Detroit who buy that argument - I don't think they're going to be persuaded."
Romney has argued that had the companies been pushed into managed bankruptcy without government assistance, they would have restructured and returned to profitability more quickly than they have.
ABC News' Simone Press contributed to this report.