I used to believe that "business" could do whatever it wants on private property, but after a lot of thought, I have decided that is an anarchistic position that has no place in civilized society.
First, it is legal for the state, via power given to it by the people, to regulate business and commerce. The US Constitution does not forbid this power to the states and every state constitution in some way creates laws and statutes regulating business.
When you open a business "TO THE PUBLIC" with no controlled access and with the intention of providing goods to the public, it is not a unilateral ultimatum, but rather an agreement between the business owner AND the public as to the terms of how that commerce will occur.
For example, the public gets to decide how many rats you have in your freezer if you run a restaurant (it better be zero) and the public gets to decide that you call a "gallon" of gas the same thing the rest of the public calls a gallon of gas. The public also tells business owners to collect sales tax, safety and occupancy laws and a myriad of other requirements that exist in order to do business with the public. Oh and they get to decide that your employees MUST WASH HAND BEFORE RETURNING TO WORK.
In general, I am for these regulations as they allow the public to buy and sell safely and efficiently.
As a society, we have decided that discrimination by business is not desired. As a white, middle aged guy who has never been turned away from a restaurant or hotel for being the "wrong color" or wrong religion, I agree with public accommodation as it exists in the US. Renting a room to a Jew, or selling gas to a black person is NOT enough of a violation of your right to free association to allow it to trump the rights of all in the US to be treated equally with respect to commerce.
Therefore, I think business should have no more right to refuse those of the PUBLIC that carry concealed than they do blacks or Baptists or Bill who is wearing his wife's underwear because he "likes how it feels." What you believe, your sexuality or what is under your clothing is, quite frankly, none of the shop keepers business. If a person has no interest in agreeing to the terms the public has created, then that person has no business opening their doors.