Not really likely, right?
So logically, the agent had to be informed that the passenger was making his angry tweet, which, as you imply, the passenger was actually using the tweet to blackmail the agent into bending the rules for him.
I think it is very key, and very telling that this is not addressed in the story. There is no way for Southwest employees at that gate to have known this guy tweeted anything, without the passenger informing of it, and once we get to that obvious fact, to what end was he doing this? The obvious reason is to intimidate the agent he felt was "rude" - which seems rather petulant.
Now it gets more interesting if you start to wonder if there was a reason why the agent threatened to call the police on this guy... was it an overreaction, or was this passenger just being such an incredibly overbearing, pompous ass in his blackmail attempt, that the agent felt threatened? It might be that they never requested him to remove the tweet, but were instead responding to his petulant tantrum.
I can easily see it playing out that he was informed he needed to calm down and back off or they would call security. We only have his word that they threatened to have him arrested, and that he had to remove the tweet... it seems more reasonable, knowing this passenger intended to intimidate the agents, that he was in a threatening posture, and realizing he was about to get a royal TSA probing, "calmed down" and offered to remove his tweet as a gesture - all the while plotting to tell the story we see presented here, in all of its one-sided glory.
I hate to side either way on this story, but I'm more inclined, given this key missing item of the story, to believe that the "more to this story" involved the passenger being a LOT more in the wrong than the gate agent.