In Indiana, the fight is particularly complicated for Republicans, who find themselves tugged between two powerful groups in their party that are at odds on the issue: business interests and social conservatives.

Though businesses are not of one mind, a significant number of corporate leaders say Indiana needs to embrace openness toward gay people if it wants to compete for skilled workers. Now is a good time to do so, those leaders say, to help repair damage to the state’s reputation after the fight over the religious freedom law.

Under pressure from some of the state’s leading corporations, state lawmakers added language to the new law in April explicitly stating that it does not authorize discrimination against gay men and lesbians, yet some said the entire episode pointed to a dearth of protections in Indiana. This month, William S. Oesterle, a Republican and the former chief executive of Angie’s List, announced that he was starting a coalition of tech sector businesses intent on pressing lesbian, gay, bisexual and transgender rights across the state.

But religious conservatives are resisting. “This is the war within the Republican Party,” said Curt Smith, the president of the Indiana Family Institute, one of several prominent conservative groups in the state that oppose such measures. “They are equating a religious view of marriage and not wanting to be involved with the celebration of something else with racism and bigotry, and that’s not right.”

As for an envisioned statewide anti-discrimination measure, Mr. Pence, who is up for re-election to a second term in 2016, has not taken a position. The governor “is listening to people on all sides of the issue in order to determine how best to move forward,” his spokeswoman, Kara Brooks, said. Some Republican lawmakers also said it was too early to comment on such a measure.

But advocates said they were confident it could pass.

“This is very simple, and Hoosiers are fair people,” said Chris Paulsen, the campaign manager of Freedom Indiana, a group that fought the proposed ban on same-sex marriage in the state and is now pressing for a statewide anti-discrimination measure with its organizers, email list and money from national donors.