'Although Elane Photography made a religious liberty argument in the lower court, they did not bring this claim to the Supreme Court. Instead, they argued that complying with New Mexico’s ban on anti-gay discrimination would “require them to express messages that conflict with their religious beliefs” in violation of the First Amendment’s free speech guarantees.'
This 'argument' would have failed as well, given the Employment Division v. Smith decision was unanimous, holding that religious beliefs cannot 'justify' ignoring or violating just and proper laws, and that just and proper laws, such as New Mexico's public accommodations law, don't violate the First Amendment.