The Welsh club's Malaysian owner blamed the Scot for airing "far too much dirty linen" this season, drawing focus away from the on-pitch challenges the side have faced
Mackay - who led Cardiff to the Premier League for the first time in their history - was relieved of 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, after growing frustrated with the Scot's public request to sign three players in the January transfer market.
Tan responded by criticising the pre-season dealings done by Mackay and head of recruitment Iain Moody, who also paid with his job in October.
It appeared as if manager and owner would attempt to rebuild their damaged relationship, with chairman Mehmet Dalman claiming Mackay would remain in charge, but the long-running saga was finally ended on Friday.
And Tan believes the issue had become too high-profile to ignore and is in the best interest of supporters.
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.