Axelrod later told CNN's John King that the Romney campaign "jumped on (the comment) like a raft out in the deep blue sea because they were drowning under the weight of their own problems."
Before issuing her apology Thursday, Rosen told CNN that Republicans were attacking her as part of a strategy to divert attention from policies championed by Romney that will hurt women.
"The issue that I'm focusing on is does Mitt Romney have a vision for bringing women up economically and can he himself stop referring to his wife as his economic surrogate," Rosen said.
Her apology statement said: "Let's put the faux 'war against stay-at-home moms' to rest once and for all."
"As a mom, I know that raising children is the hardest job there is. As a pundit, I know my words on CNN last night were poorly chosen," the statement said. "In response to Mitt Romney on the campaign trail referring to his wife as a better person to answer questions about women than he is, I was discussing his lack of a record on the plight of women's financial struggles."
While Romney still needs several hundred delegates to clinch the GOP nomination, Santorum's departure from the race leaves his path free of obstacles.
However, Romney's campaign still struggles to generate enthusiasm among the GOP conservative base, which questions his more moderate stances as Massachusetts governor. In addition, a Washington Post/ABC News poll this week affirmed findings of other recent polls that Romney trails Obama among female voters.
Another poll released Thursday by Quinnipiac University showed Obama leading Romney in New Jersey, even under the hypothetical scenario that popular Republican Gov. Chris Christie was on the GOP ticket as Romney's running mate.
New Jersey is considered a solidly Democratic state in presidential general elections. The Quinnipiac poll also showed a Romney-Christie ticket trailing against Obama and Vice President Joe Biden among female voters.
The Republican presidential campaign has included a conservative shift to appeal to tea party voters in the primary and caucus season. However, some socially conservative polices opposing abortion and health care coverage for contraception appear to be raising concerns among female voters.
Democrats have seized on that dynamic by emphasizing Republican stances that they say harm equal treatment and opportunity for women. In response, the Romney campaign is targeting Obama's economic policies as being bad for women.
The "war over women" erupted in full force Wednesday, when Romney said Obama may not have started the recession but his policies extended it, which hurt women.
"He just made it worse, and made it last longer," Romney said. "And because it lasted longer, more and more women lost jobs, such that in his 3½ years, 92.3% of the people who lost jobs have been women. His failures have hurt women."
An analysis of federal labor statistics shows that the Romney claim is technically true, but lacks important context.
The number of nonfarm-employed women from January 2009, when Obama took office, to March 2012 fell far more than the number of employed men in that period. The total job loss for the period for both men and women combined was 740,000. The number of women who lost nonfarm jobs in that time was 683,000, according to figures from the Bureau of Labor Statistics.
That amounts to 92.3% -- the figure Romney cited. However, the statistic does not reflect that men constituted a much larger chunk of the job loss pie in the year leading up to Obama's inauguration.
In the 2008 calendar year, men lost a total of 2.7 million nonfarm jobs, compared with 895,000 lost for women. Men made up 75.4% of the 3.6 million jobs lost that year.
Romney's claim also does not reflect that the job losses for women began in March 2008, almost a full year before Obama took office. At that point, women held a total of 67.3 million nonfarm payroll jobs, the highest level of female employment of the Bush administration.
Romney got a boost Thursday when two prominent anti-abortion groups -- the Susan B. Anthony List and National Right to Life -- endorsed his nomination, a sign that conservatives will coalesce around his candidacy despite concerns about his more moderate history as Massachusetts governor.
An influential Republican, former Mississippi Gov. Haley Barbour, said Thursday that Romney should aggressively focus on a policy-heavy message that questions Obama's handling of the economy.
That theme will be enough to keep the conservative base excited while also appealing to independent voters who will ultimately decide the election, Barbour said.
"This campaign is going to be waged in the center," Barbour said. "Don't ever forget that Barack Obama is the great uniter of Republicans. The party apparatus, the conservatives, the tea party, the organizations like small businesses and the NRA, they will be very active in providing volunteers and our base will be very active. That's critically important. But the election is going to be decided by a few million people, most of whom voted for Obama last time but have the same level of buyer's remorse as ticket purchasers on the Titanic."
Obama had no public events scheduled Thursday, but Biden used a New Hampshire campaign speech to attack Romney's economic policies, calling them a repeat of failed measures that caused economic collapse in 2008.
"Just look at what Gov. Romney wants to do," Biden said to a crowd of 400 supporters in Exeter, considered the birthplace of the Republican Party. "The Bush tax cuts for the wealthy, the ones that were intended to expire, the ones that will expire this December -- he wants to extend them."
At one point, when a baby started crying during his critique of Romney's plan, Biden quickly quipped, "I don't blame the baby for crying," which was met with applause from the audience.