Yes. Free speech only applies to government censorship, not what private individuals or institutions choose to use to filter those they care to listen to, assoc

Yes. Free speech only applies to government censorship, not what private individuals or institutions choose to use to filter those they care to listen to, associate with, or otherwise give the time of day to. Likewise, privacy isn't really a concern here, since it isn't like the organization broke into their accounts to check up on their privately held conversations or beliefs. It was posted into a group with many other loosely associated members, and was brought to the organization's attention. In your gum spitting example, it's like spitting out gum in your back yard behind a privacy fence versus spitting out gum in your employer's parking lot and having your boss step in it moments later. It's not private, and it may have consequences.