Instead, the former Massachusetts governor said states should make their own abortion laws.
"I would like to see each state be able to make its own law with regard to abortion," Romney said after a speech to about 50 small business leaders. "I think the Roe v. Wade one-size-fits-all approach is wrong."
Although Romney's presidential campaign reported earning nearly as much as the campaigns of Sens. Hillary Clinton and Barack Obama, his name recognition isn't on the same level.
A new CBS Newsthat 7 in 10 Republican primary voters say they don't really know Romney. And those who do associate one thing more than any other with him — not that he was Massachusetts governor or head of the Salt Lake Olympics, CBS News chief White House correspondent Jim Axelrod reports. They know Romney is a Mormon.
And after Islam, the Mormon faith is the one most Americans still consider a deal-breaker in backing a candidate.
But, as Axelrod reports, the candidate has faith that his faith won't be an issue.
South Carolina is an early voting state dominated by conservatives who oppose abortion rights. The ultrasound requirement has generated much emotional debate in the state Legislature.
Proponents think women would change their minds after seeing an ultrasound and choose to keep the child or offer it for adoption. Critics consider it a way to intimidate women who already have made a difficult decision.
The House approved a version of the bill requiring women to view the images. If that proposal were enacted, it would make South Carolina the only state with such a requirement. However, a state Senate panel this week removed that language, setting up a possible showdown between the two Republican-dominated chambers.
South Carolina Attorney General Henry McMaster, a Republican, said he supports the requirement, but forcing a woman to view an ultrasound would be illegal. McMaster's opinion led the senate subcommittee to make viewing the ultrasound an option, rather than a requirement.
Critics have said Romney has reversed his position on abortion rights, and he has sought to reassure social conservatives that he sincerely opposes abortion.
Kansas Sen. Sam Brownback, also campaigning in the state, reaffirmed his support Thursday for legislation that would mandate the requirement.
Romney's visit began a two-day swing through South Carolina. He plans to visit several county conventions, which his campaign has called key building blocks for the primary race here.
Also Friday, Romney announced that Lt. Gen. Claudius "Bud" Elmer Watts III and Brig. Gen. Thomas R. Mikolajcik would serve as chairmen of a veterans coalition supporting Romney in South Carolina.
The announcement came one day after South Carolina Adjutant Gen. Stan Spears threw his support behind Arizona Sen. John McCain, announcing he would lead a similar veterans panel.