Michael Colgan has apologised to any person who he has hurt or who was made to feel upset.
He says that through "misjudged behaviour" he caused upset to some of his co-workers.
Mr Colgan writes: "I already knew that I was not politically correct, that I often sacrificed proper conduct for a punchline, and that, at times, could be too exacting as a boss.
"But realising that I have been responsible for causing distress to some of those with whom I worked so closely has shocked me, and I am truly sorry."
He added that his behaviour should not be equated with sexual crimes, and writes that his family has "had to suffer totally false suggestions that I might be guilty of more than misjudged behaviour".
He expressed concern that "one can be put through such a public online trial with the media as judge and Twitter as jury".
Mr Colgan apologised to "any person, in or out of the office, whom I have hurt", and noted that the last thing he wanted was to distract from the 'excellent work' of the new team at The Gate.
On Thursday, the theatre announced details of an independent review into the allegations against Mr Colgan.