If he left the tweet online and walked out of the airport he wouldn't get arrested. The tweet isn't the issue, his persistent demands that he's let onto the plane would be the issue.
He wasn't given the choice to walk out of the airport. He told to a) delete the tweet and reboard, or b) go to jail. There no was option c to leave, and he even if he had, he wouldn't have been refunded for the unused tickets.
Arresting someone based on something they said or tweeted (that wasn't in any any threatening) *is* a first amendment violation. I'm sure you're right, though, they would have called it a public disturbance or some such. How complaining calmly about poor service is a disturbance or threating is another question. In general, airline employees are given too much latitude to be a*holes in the name of protecting safety.