But religious freedom advocates argue the rights of same-sex couples shouldn't supersede an individual's First Amendment right to religious freedom. If a vegan baker isn't required to sell non-vegan items that go against their ethics, why should a Christian baker have to sell same-sex cakes that go against their morals?
Conservatives say people should not be compelled to act against their beliefs. As the baker saw it, creating a wedding cake for the gay couple means tacitly approving of gay marriage—his talents, which he sees as God-given, are being used to create a special celebratory cake originally meant to celebrate the union of husband and wife. It comes down to the First Amendment; the baker sees his cakes as a form of self-expression and he's entitled to practice his religion without government interference.
A private business should be able to run their business in accordance with their faith. This was upheld by the Supreme Court in Burwell v. Hobby Lobby Stores, Inc. where the court found in favor of Hobby Lobby, saying the company has the right to exempt itself from providing birth control to its employees because of the company owner's religious beliefs. The court found the government cannot compel Hobby Lobby to provide birth control to its employees because it would violate the employer's religious beliefs.
In a similar vein, business owners should not be forced to participate in something that violates their religious beliefs. This isn't discrimination—it's religious freedom.
Thomas Kidd argues in The Gospel Coalition:
Phillips says that baking a cake is an artistic expression subject to First Amendment free speech protections. Phillips has provided bakery services to gays under other circumstances, so his point is not that he won’t serve gay customers. It is that he objects to gay marriage and does not believe that the state should force him to create an artistic product under any circumstances, much less one that violates his conventional, traditional religious beliefs. As a matter of policy, Phillips also won’t produce Halloween cakes, or cakes that feature any profanity or suggestive themes.
As Kidd notes, the baker didn't outright refuse to make a cake for the same-sex couple, he just refused to use his artistic expression to endorse a behavior he felt violated his personal beliefs, similar to his refusing to make explicit or devil-themed cakes. Kidd seems to believe precedent in on the baker's side.
Court precedent has generally frowned upon the idea of the government forcing people to act against religious conscience. Even at the time of America’s founding, political leaders were well familiar with extending conscience exemptions to groups like the Quakers.