Earlier quoted context omitted.
Yes, provided they are not deciding on the basis of someone belonging to a protected class. Hence a cake maker can cheerfully refuse your business on the basis of you wanting a cake that says “Jennifer Garner Eats Babies” but not on the basis of you, say, being homosexual, or Black.
That means I can refuse servicing you based on, for example, the clothes you’re wearing, right? Or maybe because I don’t like your hair? As far as I know people with bad hair is not a protected class. I’m trying to figure out how do you know the basis on which some private enterprise decided to refuse to provide their service to someone. Is it based on their stated reason?
If you’re trying to understand the basis by which courts figure this stuff out, what we’re talking about here is the Civil Rights Act of 1964, which is civil law, and the evidentiary standard that applies is “preponderance of evidence”.
The businesses’ stated reason is certainly one piece of evidence that courts weigh, but it’s far from the only one. If you refuse me business based on what I’m wearing, but I can show you regular do business with people wearing the same thing, or I can demonstrate a preponderance of evidence that you knew I was homosexual and had made statements to the effect that you would refuse to serve homosexuals, courts aren’t going to just take you at your word.