-You can refuse service for conduct, such as drunk and disorderly, must be objectively identifiable, consistent standard.
-You can refuse service to people who’ve robbed you or vandalized your property, again this is specific, objectively objectionable conduct.
-You can refuse service based on dress code, if reasonable and applied equally.
-You can refuse service to everyone, equally, by closing early.
-You can refuse service to the public by simply not being open to it, by only serving certain licensed professionals, such as electrical contractors or law enforcement. But the criteria must be clear and based on objectively defensible business practices. It cannot be subjective, it cannot be rooted in bigotry or religion.

If your business is open to the public the one thing you cannot do is pick and chooose, parsing that public, refusing service based on perceived status or class. You cannot single out blacks, jews, gays, or redheads.

You are not allowed to do this because your license to do business is provided by the State, which has an obligation to make sure everyone has equal protection of the laws. Your business benefits from public roads and infrastructure, you depend on public law enforcement for dispute resolution and protection, on public fire departments for safety. You can’t subjectively ban only certain elements of the public, because all those elements helped make the environment conducive to the success of your business. It would be fundamentally unfair to demand that gays help pay for roads that you benefit from if you won’t serve them. It would be fundamentally unfair to ask them to help pay for the services of the police that then enforces your bigotry against them.

If you want to be a bigot, do it privately, in your own home. Nobody will force you to associate against your will there, no matter how dickish your will may be. When you open a business to the public you’re surrendering a certain amount of control over what that public looks like in exchange for the numerous benefits that same public provides to you.