If he used a work computer, email address or identified himself as an employee of the paper, they have every right to fire him.
From the article:
He was told that Trevor Thomas, deputy communications director of the Human Rights Campaign, had Googled his name, discovered he was a reporter, and was demanding Grard be fired.
Doesn't sound like Tevor Thomas cared about any of the three 'possible' infractions you noted (i.e., he didn't factor them into his conclusion that Mr. Grard must be fired).