Do the business owners in Arizona not have the right to refuse service to anyone of their choosing?

Any business can refuse service to a specific person, but singling out a group or groups is discrimination and may be in violation of State and Federal civil rights laws.

By the way, that Arizona bill even allowed employees, including government employees, to refuse service and not be sued.

Bflowing is correct. A business can refuse services to a person if they have a legitimate reason. For instance, If the person is drunk, a business could refuse to sell him alcohol. But a business can't discriminate against a group of people. Refusing service to gays, simply because they are gay would be a violation of their civil rights.

They have the right to refuse service to anyone and throw anyone they want off the property they own or are leasing.

Of course they do - unless the owner is Christian AND the customer is gay.

Take-out EITHER ONE of those factors and there's no problem.