Yes they can
https://en.wikipedia.org/wiki/Marsh_v._Alabama
If the private corporation acts as a state actor, meaning they are acting like a government. Otherwise the government would just “encourage” private corps to censor what they don’t like.
Granted, this is exactly what Facebook is doing, but no one has smacked them down on it yet.
“ Granted, this is exactly what Facebook is doing, but no one has smacked them down on it yet.”
Waiting for the Section 230 ruling.