Consider, a "privately" owned company that opens its doors to the public is absolutely not allowed to refuse to serve someone on the basis of their skin color.
The fact is that when you open your service to the general public then you are no longer strictly speaking a "private" organization: you open yourself up to be regulated by the governments of those people you invite into your service. When you operate what can be argued is very close to a natural monopoly, you have even more duties.