Celtic chief executive Michael Nicholson said that three of his colleagues were "assaulted" after Sunday's Hampden defeat as he declared Peter Lawwell's resignation a "sad day" for the club.
Outgoing chairman Lawwell cited "intolerable abuse and threats" as he announced the end of his 18-year spell at Celtic Park.
A number of Celtic fans had been campaigning for boardroom change after directors took responsibility for their Champions League qualification failure and inability to achieve their summer transfer window objectives.
Lawwell had come under increasing pressure following his handling of the club's abandoned annual general meeting and Wilfried Nancy's torrid start as manager saw some fans gather to protest outside Celtic Park after Sunday's Premier Sports Cup final defeat to St Mirren.
Giving his reaction to Lawwell's decision, Nicholson told Celtic TV: "I'm hugely disappointed. It's a sad day for Celtic. I think it's sad because one of our colleagues has felt the need to stand down from his role.
"Peter's a Celtic fan first and foremost, but he's been a great colleague, a great leader for Celtic, both as chief executive and more recently as chairman, as we've enjoyed great success on and off the pitch.
"But it's a sad day for Celtic because of the reasons behind Peter's decision to stand down.
"I know it's a decision that weighed heavily on him. It's not an easy decision, but I think he's made the decision for himself and for his family because of the abuse and because of the threats that were directed towards him.
"Those are behaviours that we've seen from perhaps a very small minority of the Celtic support, but those are behaviours that are completely unacceptable.
"Most recently, we had a situation where three of our colleagues were assaulted at Celtic Park on Sunday night after the match.
"As I say, these behaviours and conduct are completely unacceptable. For me, they're not Celtic. They don't sit with the values of Celtic supporters and the Celtic family.
"As a board, it strengthens our resolve to do the right thing for Celtic and to take this club forward together for the future."
Nicholson declared the board's resolve to support Nancy as "absolutely solid" hours before the club lost its fourth consecutive game, a 2-1 defeat to Dundee United.
When asked if he heard chants calling for him to leave, the 48-year-old Nancy said: "I heard it. And obviously, it's not easy to hear that. But I can understand their frustration too.
"It's new for me to have four losses in a row. This is also new for the players.
"I can understand they are upset. But the only thing that I can tell them is that I think we had a good performance.
"We are missing the goals that could have been good for us and we are still working on that to win the next game.
"We don't have a lot of time to work but when we work, we do it well. And again, that was a good performance. But the result was not there.
"We had the possibility to score many goals, because we had a lot of chances. And if we don't score, if we don't keep the game after that, the game is not over. That's why it's difficult to accept that, but this is the reality."
Information from PA was used in this report.