Pence's bind was clear from the press conference, in which his overall message was: This law is not broken, but it's imperative to fix it anyway. The question is whether the clarification will be enough. Some critics, including in the business community, worry that the damage done by the law is already too great, and that it must be fully repealed rather than simply tweaked.

Pence called for a bill on his desk by the end of the week. That deadline coincides with the NCAA Final Four, which begins Saturday in Indianapolis. The NCAA has criticized the law, and there have been calls—though they have not gained traction—for the tournament to be moved.

It's the culmination of what Pence has acknowledged as a bad week for his state. On Tuesday, The Indianapolis Star demanded changes in an unusual and striking front-page editorial. The paper's editorial board says that the state's reputation is on the line. "All of this is at risk because of a new law, the Religious Freedom Restoration Act, that no matter its original intent already has done enormous harm to our state and potentially our economic future," the editorial states.

In addition, nine CEOs, including some of the biggest employers in the state, sent a letter to Senate President Pro Tem David Long and Speaker Brian Bosma calling for changes. As it happened, Long and Bosma were already moving in that direction. "To the extent that we need to clarify through legislative action that this law does not and will not be allowed to discriminate against anyone, we will do just that," Long said at a press conference Monday.

The somewhat hedged language in Long's statement showed the way the law has split players into three broadly defined camps. One conservative group insists the protections in the law are necessary to allow citizens to practice their religion without interference, especially with marriage equality on the march across the nation. A second, on the left, insists that any law is unacceptable because it enshrines bigotry against gays.

The third group is the pivotal one. It includes conservatives and liberals alike—people who generally think it's reasonable to have laws that protect citizens' private practice of religion but also don't want laws that allow businesses to discriminate or deny service to gays and lesbians. Crucially, however, there is disagreement on whether Indiana's RFRA does that.

"Had this law been about legalizing discrimination, I would have vetoed it," Pence said. "This law does not give anyone a right to discriminate. It does not give anyone the right to deny services." He blasted the national media for what he said were misrepresentations of the law, and he pointed to the example of a federal RFRA and similar laws in 19 other states.

But as my colleague Garrett Epps pointed out Monday, Indiana's RFRA contains language in two places that sets it apart from the earlier laws, by allowing for people to claim discrimination in private business that doesn't involve the government. That seems to enable the law to be used by business owners to deny service to, say, gays and lesbians.