Nearly everyone (at least, everyone I know) thinks that no matter how you interpret Islam, it doesn't give you the right to hijack a plane and fly it into the side of a skyscraper. And almost nobody in the Christian world thinks that the Inquision or the Crusades were positive developments.
Perhaps being refused service on the basis of your sexual orientation doesn't compare to such tragedies. Still, I am guessing that almost any first-year law student or seminarian would tell you that if religion doesn't confer the right to commit murder, it also shouldn't allow discrimination.
Apparently, that's not how legislators in Arizona see it. They've passed a bill that would allow businesses and other establishments to refuse service to LGBT people.
So, for example, it would be perfectly legal for a baker to refuse to make a cake for a gay couple's wedding. Or a photographer could decide he didn't want to record a same-sex ceremony.
The two examples I've cited have actually occurred in other states. Now Governor Jan Brewer, on whose desk the bill sits, must decide whether she'll allow such things in her state. Given her record on civil rights issues, I'm not optimistic that she'll veto it.