No and no, I'm not advocating anything of the sort. My point is simply that property holders, at least in America, are under no obligation to enforce rights. They are under obligation to not commit crimes, and as a nation we've decided that a business refusing service on the basis of skin color is a crime.
Refusing to rebroadcast or support the opinions of someone you don't like is at present not a crime, and I sincerely hope it stays that way. Because otherwise you have to apply the same logic to other rights. And that sort of extremism leads to some crazy outcomes.
For example, in the US the 2nd amendment is, under the law, of equal importance as the 1st. So should I, as a gun owner, demand that I should be allowed to carry my firearm anywhere, on anyone's property; and claim that for them to deny me that right is a violation of my 2nd amendment rights? I'm pro-gun and even to me that sounds crazy, as it would to many pro-gunners who are usually big on personal property rights. There's a reason Full30.com exists.
Or how about the right to freedom of assembly (also part of the first amendment)? Should I be forced to allow protesters to use my property, as disallowing them violates their first amendment rights?