Moody, who had worked with Mackay at Watford, has been replaced by 23-year-old Kazakhstani Alisher Apsalyamov on an interim basis.
Apsalyamov is thought to be a friend of Cardiff owner Vincent Tan's son and appears to have no previous experience in football.
Friday's press conference ahead of Saturday's Premier League meeting with Chelsea was the first opportunity for Mackay to speak publicly on the issue.
But, after clarifying there had been no overspend in the summer transfer budget and that he had not been asked to resign, he refused to answer any further questions on the matter.
He said: "Iain is an absolute class act as far as I am concerned. He has done a very impressive job for the football club.
"In one of the periods of success at the football club, which we have been in over the last two and a half years, he has had a huge part to play in the restructuring of my squad.
"He helped bring in more than 20-odd players and he is someone who had a huge part to play in the current success, who will be missed by everyone who works for out football club.
"He leaves with my utmost gratitude and respect. I would say our loss will be someone's huge gain.
"One thing I would clarify, as I have read a few things over the past couple of weeks, is that we were absolutely under budget this summer. I want to clarify that. That is what I have to say about Iain.
"As far as I am concerned, again for clarification, under no circumstances was I asked to resign and at no time have I thought of resigning from the football club."