Livejournal's Terms of Service
Terms of Service. Basically the rules for posting here. While LJ is publicly accessible, since it is a privately-owned facility, it is not subject to the 1st Amendment, and therefore anyone can be banned from posting here.

Therefore, claiming your 1st Amendment rights are being stifled by being banned, doesn't apply here. It would be like claiming you have the right to enter Toys-R-Us and scream obscenities. Actually Toys-R-Us has the right to kick you out and post a guard to keep you out, for as long as they deem necessary. You have no say, and neither do the courts. You could possibly argue that they are being discriminatory, which in places like California is a valid argument, but you would need to prove that they are basing their ban on race, sex, sexual orientation, etc, and not just on the fact that you are acting like an ass.