Pence defended the measure as a vehicle to protect religious liberty but said he has been meeting with lawmakers "around the clock" to address concerns that it would allow businesses to deny services to gay customers.
The governor said he does not believe "for a minute" that lawmakers intended "to create a license to discriminate."
"It certainly wasn't my intent," said Pence, who signed the law last week.
But, he said, he "can appreciate that that's become the perception, not just here in Indiana but all across the country. We need to confront that."
The law prohibits state laws that "substantially burden" a person's ability to follow his or her religious beliefs. The definition of "person" includes religious institutions, businesses and associations.
Although the legal language does not specifically mention gays and lesbians, critics say the law is designed to protect businesses and individuals who do not want to serve gays and lesbians, such as florists or caterers who might be hired for a same-sex wedding.
Pence said the measure has been mischaracterized.
“This law does not give anyone a license to deny services to gay and lesbian couples," he said.
Businesses and organizations including Apple and the NCAA have voiced concern over the effect of the law, and some states have barred government-funded travel to Indiana.
Also Tuesday, the Indianapolis Star urged Indiana lawmakers in a front-page editorial to respond to widespread criticism of the law by protecting the rights of gays and lesbians.
The newspaper says the uproar sparked by the law has "done enormous harm" to the state and potentially to its economic future.
Meanwhile, Arkansas was poised to follow Indiana in enacting a law despite increasing criticism from businesses and gay-rights advocates.
The Arkansas House could vote as early as Tuesday on a proposal that would prohibit state and local governments from infringing on a person's religious beliefs without a "compelling" reason. And unlike in Indiana, Arkansas lawmakers said they won't modify their measure.
"There's not really any place to make any changes now," Republican Rep. Bob Ballinger of Hindsville said about his proposal. "If there are questions in two years, we can fix it."
Hundreds of protesters filled Arkansas' Capitol to oppose the measure, holding signs that read "Discrimination is not a Christian Value" and "Discrimination is a Disease," and chanting "Shame on You" at Ballinger after the measure was endorsed by a House committee.
"I believe that many people will want to flee the state, and many people will want to avoid our state," said Rita Jernigan, a protester and one of the lead plaintiffs in a federal lawsuit challenging Arkansas' gay marriage ban. IBJ.com