The governor's statement came less than a week after he signed the measure into law. It created a furor, drawing protest from CEOs including Apple's Tim Cook and even the Indianapolis-based NCAA, whose Final Four men's basketball championship begins this weekend in Indianapolis.
Read MoreNCAA president: Indiana law against our core values
Separately on Tuesday afternoon, New York Gov. Andrew Cuomo banned "non-essential" state-funded travel to Indiana. The ban would take effect immediately and bar trips not vital to law enforcement or public safety, he said.
"With this action, we stand by our LBGT family members, friends and colleagues to ensure that their rights are respected," Cuomo said in a release.
University of Southern California athletic director Pat Haden tweeted that he will boycott a College Football Playoff committee meeting in Indiana.
Critics said the law, which is to take effect July 1, allowed for discrimination against gays and lesbians under the guise of protecting religious beliefs.
Pence admitted that what he is calling for was both a clarification and a "fix."
"This law does not give anyone a license to deny services to gay and lesbian couples," the governor said, saying that any addition would be consistent with what the state's General Assembly had intended.
Still, a change would help clarify that intention, he said.
"I think it would be helpful, and I'd like to see on my desk by the end of this week legislation that is added to the Religious Freedom Restoration Act in Indiana that makes it clear that this law does not give businesses the right to deny services to anyone," he said.
Billionaire investor Warren Buffett told CNBC on Tuesday that he would oppose the Indiana law if it ended up leading to discrimination against gays and lesbians.
"If that's a possibility, I'd say they'd better get rid of it," Buffett said from the J.D. Power Automotive Forum.
Some noted Tuesday that the governor's call for legislation clarifying that the state is not giving businesses the right to discriminate, is not necessarily the same thing as a new law offering protection to certain groups.
"We will review whatever changes come across with an open mind but we've got a serious perception problem in the state that's impacting our state and our city, and any solution has got to remove all doubt that individuals can be discriminated against or it's just not going to work," said Michael Huber, CEO of the Indianapolis Chamber of Commerce.
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"Does the clarifying language that says businesses may not discriminate, could that serve as some kind of a substitute for stronger anti-discrimination laws? I don't know," Huber added.
At the news conference, the governor said he was "taken aback" by the backlash against the law, blaming "sloppy reporting" and "grossly mischaracterized" descriptions of the bill for the uproar.
Not everyone agreed, however, with the governor's depiction of recent events.
"When [Pence] says the bill has been mischaracterized, it has been mischaracterized up to an extent, but parts of it have been accurately characterized as well," Huber said.
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The speaker of the Indiana House, Brian Bosma, pledged to fix the law by making it clear that it does not condone discrimination.
"The bill does no such thing to allow discrimination," he told CNBC's "Squawk on the Street" on Tuesday. "We have to address the perception and here is my pledge—we're going to have a legislative fix that makes it crystal clear that gays and lesbians cannot be discriminated against in Indiana."
In an op-ed in USA Today on Tuesday, Bosma sought to assure the public that the law would include language that makes the intent clear that no member of the general public can be denied goods or services.
"We need to get that misperception off the books entirely, and that's what we're working on in a very fast fashion," he told CNBC. "The best thing to do here is to fix it, draw that discussion to a close and move forward."