I could be wrong, but...
The Civil Rights Act of 1964 rightly prohibited discrimination on the parts of companies that provide services (such as restaurants and hotels) based on things like race, creed, or ethnicity.
Federal law also rightly prohibits discrimination on the bases of such things as gender or age. (Thought such discrimination remains and is often difficult to prove.)
All of these prohibitions, in other words, ban discrimination or refusal to serve clientele on the basis of classifications.
And that is as it should be. Every American, including those who identify themselves as LGBT, should be protected against discrimination.
But the RFRA law in Indiana appears to have been about an entirely different subject altogether.
It was meant to ensure that businesspeople who have moral objections to gay marriage did not have to violate their consciences by being part of such a procedure.
People may disagree with such moral scruples. But just as I would hope that we would recognize the conscientious objection to war of someone seeking exemption from military service or the legitimate refusal of a businessperson not wanting to do business with the Ku Klux Klan, I also hope that we would recognize the right of conscientious objection to gay marriage. In America, no one should be forced by law to violate their conscience either in their personal or business lives.
Discrimination based on gender identification is immoral.
Commanding people to violate their consciences is immoral too. And deeply Un-American.
While admittedly I'm unaware of all the stipulations of RFRA, I cannot see how protecting people's right to exercise conscience threatens anyone's civil rights.