Mackay - who led Cardiff City to the Premier League for the first time in their history - was relieved on his duties, following weeks of speculation he would depart the club.
The relationship between the two became strained when the Malaysian reportedly issued an ultimatum telling Mackay to resign or be sacked, having taken issue with the Scot's public request to sign three players in the January transfer market.
In a statement by Tan reported by walesonline, he said: "There has been a good deal of publicity generated by and about Mr Malky Mackay over the last few months.
"Indeed far too much dirty linen has been exposed to the public gaze. But, I stress, not by me.
"Indeed, I have deliberately not responded to this, hoping that the club can be judged on its football rather than personalised arguments about who said what to whom.
"I have, however, regretfully concluded that it is no longer fair to the club, its players, its fans, or the public more generally, for this uncomfortable state of affairs to continue. Cardiff City Football Club means far too much to us all for it to be distracted by this."
Cardiff's first game in the post-Mackay era is a home fixture against Sunderland, with Mackay's former assistant David Kerslake and first team coach Joe McBride reportedly set to take temporary charge.
