In essence, yes.
People keep throwing around this term "public accommodating businesses" with the goal of trying to make someone's private business part of the public sector in order to give control to the government. However, that's simply not the case.
If the government or the public didn't put up any money to start that business, then it's still private property.
I feel the same way about this as I do about those silly laws that allow your neighbors to dictate what you can and can't have in your yard (with government force). In many localities, if your neighbors think that you have something in your yard that makes it look trashy, they can petition the city and the city can make you get it out of your yard, even though the neighbors aren't making the mortgage payment...
In essence then, would it be accurate to assume you believe the civil rights act in also unconstitutional?
Private property really has nothing to do with this, virtually all businesses are private property, which the property owners chooses to open up to the public at large. They can have a private business, if they choose to affiliate with a church to get customers or became a private club and did not cater to the public. Obviously, they don't want to do that, since they want the advantage of capitalizing on the numbers in the public, but still be allowed to refuse service to some.
As I have stated many times, I would have no problem with businesses being able to refuse service to whomever they chose to, as long as they declare publically who the people are they are going to refuse service to, since they are open to the public. Not so sure many would want to do that though.
In regards to what you can or can't do with your private residential property, if you live in a subdivision with a home owners associate, the courts have held you can be restricted what you do with your property if three tests are met:
-It is a safety issue that impacts others in the neighborhood
-It impacts protected land - wet lands, tree preservation areas etc..
-It negatively impacts others property values; un kept land, home in disrepair, junks in the driveway, etc etc.