Not the way I'm using, and it doesn't even have to do with social media. A TV station doesn't have to show "all content", and a local Alcoholics Anonymous doesn't have to accept someone who keeps telling others to drink alcohol.What if the top TV manufacturer started to use their position to interfere with the viewing of "liberal" content? What if there were other TV manufacturers, but the others only accounted for less than 10% of the supply, and due to network effects and exclusivity deals, it was impossible for anyone to make a living making content for the other 10% of TVs? Monopolies are bad, and what constitutes a monopoly changes with technology.
But that right doesn't mean that people can't show you the door from their private space.
Sure. But how far should "showing you the door" go? It seems to me that the ethos of many of us in Bay Area companies would be to segregate those they would classify as wrong-thinkers to less desirable neighborhoods, kick them out of the choicest avenues for commerce and networking, and to deny them the best that society can provide. That's basically the same kind of behavior that "mainstream" people used to apply towards homosexuals, black people, people of my own ethnic group, the polyamorous, pagans, Jewish people, etc. The left rightfully points out the injustice of marginalizing people from public life based on inherent characteristics. The left also used to call out marginalizing people based on their private life choices that affect no-one. But when it comes to politics, all of a sudden, they are right and empowered to basically do the same thing, regarding politics, "for justice?" No, that stinks to high heaven like the corruption that power brings.
So yes, private parties are allowed to do that with their own property. But the long arc of history shows, that it's the magnanimous people who are the harbingers of the future and a better world. It's the people who get power then decide it's time to wreak revenge who turn out to be the villains. In the end, it's the better way of doing things which wins out over coercion. In the end, truth will come out, and the better way will sell itself.
I could see an argument saying that social networks and YT are equivalent to a public square or any other public space.
Also note that what constitutes "private" has changed, based on technology. It used to be that the sky over your property belonged to you for infinite distance. Then a farmer tried to get a court to make all of the airplane operators pay him a toll, and the law changed to accommodate progress. What if, in the early days of telephone, the operators decided that people of your sexual orientation should be "shown the door," and they started to make it harder for you to call or for people to call you based on how you used their private network? What if they gave a special ring only to those with orientations they approved of, and a different ring to people they didn't approve of? None of that would strike me as at all fair and just, but prior to wiretap laws, your argument would have justified all of that. Wiretaps are now illegal, but wire-based communications existed before the wiretap laws were written. The phone network is now something somewhat public that all parties have legal access to. Technology changes the social landscape, which necessitates changes in the law.
In the end, it's the live and let live society that most quickly finds the truth, makes the most money, and generates the most human happiness and progress.