"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."


That means not being forced to accept something against your beliefs that you know to be wrong. Such as a twisting of the definition of marriage that has been clear for the entire existence of human civilization.
You can have gay benefits in a secularist framework (not intruding on religious beliefs or freedoms) by calling it civil union, but don't twist the definition of marriage to be something other than what it truly is, and has been for 6,000 years.

Kim Davis a County Clerk in Kentucky has every right to oppose that, and gay Nazis are just looking like the spiteful intolerant jerks they are by imprisoning her, and trying to make her compromise her beliefs. It is just making her a martyr, expanding her support nationwide.

There are similarities and differences between the two cases.

Kim Davis is employed by the state of Kentucky, and therefore has greater leeway under the Constitutional/First Amendment protections of law.
Charee Stanley is an employee of ExpressJet Airlines, and is subject to the job requirements of her private employer.

I'd also argue that Islam is inherently in opposition to the U.S. Constitution, and that it should therefore not be protected by the Constitution, any more than Stalinism, Nazism or other insurrectionist insurgent movements should be protected. Although the unfortunate reality is, they probably would be given the same and equal protection under U.S. law. And under the current liberal/secular dominance of the courts, a muslim's protected freedoms would probably be given more consideration.
If the airline flies outside the United States, I'm not sure the same U.S. First Amendment Islamic/religious freedoms would apply outside the United States.

To be Constitutionally protected, the person should be committed to the United States and to preserve, protect and defend its Constitution in the first place. The fact that CAIR (an Islamic front group) is pushing the case, makes clear the insurrectionist nature of the suit.