With reasonable exceptions, private businesses and institutions can basically do what they want to do.
The problem, imo, is that many private Christian establishments probably don’t know the Constitution, particularly Section 1 of the 14th Amendment, a short paragraph, well enough to protect themselves from unconstitutional legal attacks by low-information pro-gay activist state and government officials including judges.
I’ve seen how “private businesses can do what they want” with the successful suits and closures of Christian businesses which declined to participate in specifically sodomite “celebrations”.