Andrew Flanagan said he “greatly regretted” his treatment of Moi Ali, who has accused him of bullying and effectively forcing her out of the organisation.
But Mr Flanagan said he would not resign his own position, believing he has “got more right than wrong” during his tenure.
The SPA chairman was appearing before Holyrood’s justice sub committee on policing after MSPs on another committee wrote to justice minister Michael Matheson expressing “serious concerns” at the running of the organisation.
Mr Flanagan said it had been a “misjudgement” to send a letter to Ms Ali taking issue with her decision to criticise the holding of committee meetings in private.
He also confirmed an SPA U-turn on the issue, saying meetings would once again be held in public, with board papers published in advance.
He said: “I greatly regret the timing, tone and content of my initial letter to her. It was a misjudgement to send a letter rather than open up a conversation. And I bitterly regret that I was
subsequently unable to allay her concerns so she could continue as a board colleague.
“She was right in raising the substantive concerns she had about transparency and perception, and she did so in a manner that was entirely consistent with her role as a public board
member. I was wrong and it is important that I today set the public record straight on that. I have now written to her and offered my full and unreserved personal apology.”
Ms Ali said she had received an electronic copy of the letter at 11.38am – just over an hour before Mr Flanagan appeared before MSPs.