This discussion is about the conflict between religious expression and accommodation laws
Just a few observations:
Almost nothing in the Constitution prohibits discrimination by private persons, i.e. private individuals and private corporations. The public accommodations part of the Civil Rights Act of 1964 was forced to rely on that catchall, the Commerce Clause. The Supreme Court upheld this law in Katzenbach v. McClung (the Ollie's Barbecue Case) and Heart of Atlanta Motel, both of which involved discrimination against blacks. The rationale was that blacks would be discouraged from taking vacations by car or otherwise traveling between states, if they could not be sure in advance of having places en route to eat, sleep, see a movie, gas up the car, and so on.
None of the three most important cases the Supreme Court has decided which involved state public accommodations laws--Roberts v. U.S. Jaycees, Hurley v. Irish-American GLIB Assn. of Boston, and Boy Scouts of America v. Dale--involved religious freedom. Roberts and Dale were First Amendment freedom of association cases, and Hurley was a freedom of speech case.
All three are a good source of information about the constitutional issues involved. Hurley is interesting because it touches on government-compelled speech, a subject the Court discussed in Barnette, Wooley, Pruneyard Shopping Center, and a few other decisions. The freedom of speech includes the freedom NOT to speak, and government violates the freedom of speech when it compels people to express ideas they do not agree with. A Jewish baker might have to bake an ordinary cake for the local Nazi group's celebration of Hitler's birthday if they ordered one, but no public accommodation law can require him to decorate it with swastikas and the phrases "Sieg Heil" and "Judenetum ist Verbrechertum."
In common law, the main types of public accommodations required to take all comers were innkeepers and common carriers. The reason was simple. Inns often were the only place for miles around to get shelter and a meal, and in bad weather, turning a person away who could have been accommodated could seriously threaten his health or even his life. Same with coaches and trains--a person might have an urgent need to get someplace, maybe for medical treatment. The idea was that by holding himself out as a provider of certain services, a private person running a business created a public expectation that he would provide that service if at all possible. (This same reasoning has applied in cases involving the duty of private water companies to contract with customers in the areas they have held themselves out as serving.)
In both Hurley and Dale, part of the problem with the state public accommodation law was that it reached too far. The "public accommodation" in Hurely was the Boston St. Patrick's Day Parade!, and in Dale it was a local Boy Scout council. That's one hell of a long way from inns, trains, barbecue joints, or motels.
For First Amendment purposes, "speech" is very broadly defined, and it includes various forms of creative expression (e.g. topless dancing). As a general rule, the more creative the speech, the more likely a state public accommodations law that compels it will be unconstitutional on free speech grounds. That's what makes the Elaine's Photography case so interesting. There was some creativity involved in Elaine's wedding photographs, and yet the work she did was not as obviously creative as, say, writing a celebratory poem or article about a same-sex wedding, or painting the couple's portrait to commemorate the occasion. I think the New Mexico Supreme Court blew that decision, misreading or ignoring U.S. Supreme Court decisions that were relevant--particularly Wooley v. Maynard.
