She shouldn't apologize, although I hope this serves as a good lesson for her, and she attempts to sound a little more intelligent in her public commentary in the future.
What really bugs the crap out of me is the part where a politician's staff 'monitors social media for mention of his name', and then feels obligated to track down the commenter. It should have been immediately apparent that the tweet was nothing particularly noteworthy, and did not warrant any further investigation. I think the way everyone behaved, from the Governer's staff on down to the Principal, is pretty damn reprehensible, and is a very frightening reflection of the degradation of liberties in this country.
Now, if a constituent was tweeting about an actual issue that the Governer could address, I can understand contacting the tweeter. Anything else just screams of police state.