I'm not too sure that i agree that a store open to the public is the same as a public place, such as the sidewalk or a park.
It's not, but it's also not the same as private property.
A homeowner can say they don't want any people of (insert race/reigion/sexual orientation) on their property, and no one can do anything about it. A business that is open to the public cannot. In this respect, it is public property
But a business that is open to the public
can dictate that a group of protestors may not assemble on his property. For these purposes, it is private property.
Businesses are required by law to accomodate certain things. By being open to the public, they give up the right to refuse service to anyone (despite those silly signs, which won't hold up in court if the person they try to deny entry to is of a "protected class", for lack of a better term).
No homeowner has to have wheelchair access to their home with steps, but a business, even on private property, must be able to accomodate a handicapped person. If accomodating them means they need to enter the building, the business
MUST install a wheelchair ramp or lift.
It gets a little trickier with a home based business, like mine that is on my own personal property. In my case, it's really about business hours. During business hours, I have to treat my property (not my home, but the shop and drive) as an open to the public business. That means people may park their car, no matter what an ugly peice of ghetto crap it is, in my drive, and I have to tolerate it. They can hang out on the property and peacefully go about their business, and I'm required to let them as long as their is no disruptive behavior. Basically, as long as what they are doing or possessing is legal and not bothersome to my other customers, it is their right. A muslim can roll out his little mat and pray in my drive, and I just have to suck it up, because even though I disapprove of the religion, I do not have the right to tell him he can't or that he must leave, so long as it's during business hours and on a part of the property that is dedicated to the business.