And that's where the issue is confused. When someone can not refuse to participate in someone else's gay or straight marriage ceremony because it violates their religious beliefs - be they anyone from an Orthodox Jew refusing to materially participate in a same-sex wedding or an Atheist refusing to participate in a religious wedding - without being deprived of their livelihood (their business) by the state then we aren't living in a free society.
No one is being deprived of their livelihood. What is being restricted is one aspect of how they conduct it. Laws against discrimination are just like regulations that tell you how much tax you pay, minimum wages, control how you dispose of hazardous waste, how many parking spaces and bathrooms you must have, whether yr employees must be bonded, what annual inspections and insurance premiums you pay, whether you need to have a state license to be in that business. Your legally enforced conformity to them is part of the cost you pay for having a business open to the public.
Under American law, the general rule is that, if you are going to have a business open to the general public, you do not have the right to discriminate merely based on a customer's class. Once you start saying "everybody but ________ general class of persons is welcome", you run afoul of our system of laws.
The Orthodox Rabbi can chose not to perform a
Jewish same sex wedding legally because his faith does not recognize the sanctity of such ceremonies. He CANNOT make the same distinction if he is open to performing the same ceremonies for non-Jews.
And an Atheist empowered to perform a wedding would be similarly restricted.
IOW, the businessman's choice is:
1) have a business open to the public and don't discriminate at all
2) have a private business and provide services ONLY to an extremely narrow clientele.
The law in question limits what the government can do.
...in a way that enshrines the beliefs of a certain set of religious persons into law, in violation of the Establishment Clause.
The law is designed so that if subsets of the state- like a county or city- decide to add non-discrimination laws, they cannot. Even if such measures had 100% approval.
The government can only step in to mediate when the conflict between someone's religious beliefs ("I won't do X") and another's commercial interests ("I want you to do X for me") reaches the point where someone trying to acquire a good or service is being substantively pushed out of a market. If nobody in town will photograph your wedding because its two men or two women you have a case for government intervention under that law. If the 3 photographers in town will take the business then you can't use the court system to troll the 1 old church lady who won't because "she's a bigot."
When the courts told the all-white Krewes of NOLA to integrate or stop parading, most stopped parading.
When the KKK had some members try to join the Krewe of Zulu- the first, biggest and still one of the only Krewes founded by blacks- they were admitted. (They don't participate much, though.)
When people try and employ law as a crude bludgeon to punish people who disagree with them, they aren't supporting Liberty but rather the exact opposite. There is a huge difference between the function of law that a hate-group like HRC wants (the ability to specifically target and punish their "enemies" for wrong-think in court) and the protections that the law is actually supposed to provide (the ability to find remedy so consumers aren't denied access to the marketplace).
If you want to do business with the public, you have to conform your business to the laws of the land.
If someone wants to make it a policy that their bar doesn't serve gays or straights or whatever craziness they want to come up with in this day and age they'll just get boycotted and market competitors will gain the benefit. Nobody is going to lose access to the market, and if they do the law gives them redress.
See my previous posts about the inefficiency and slowness of depending on politicians and the invisible hand in the arena of civil rights.
What the HRC is worried about is that without a government bludgeon they may not be able to shut down everyone they hate. The bakery of a lady who is willing to sell anybody of any sexual preference a birthday cake, but simply will not cater a same-sex wedding because such a thing is forbidden in her faith and sends the business across the street won't be fined or sued.
In what religious text is it forbidden to do business with gays? That it is OK to cherrypick which of your services you will offer them? How is this kind of discrimination justified to override that faith's version of The Golden Rule?
(Dating sites already lost on those grounds, you know.)