The Scot was eventually dismissed on December 27 after plenty of speculation surrounding his position and reports that owner Vincent Tan issued Mackay with an ultimatum to resign or be sacked.
Mackay took Cardiff into the Premier League last season but was replaced by Ole Gunnar Solskjaer early in January, and the club have since been relegated back to the Championship.
Mackay has now reached an agreement with the Welsh club over his departure, with the 42-year-old saying it offered both him and the club a "clean break".
"Today I have reached a settlement agreement dropping all claims I have made against Cardiff City Football Club," he said in a statement.
"I did not want to be in litigation and believe that it is in the best interest of all parties to have a clean break and move on.
"I have enjoyed my time at Cardiff City and am most grateful to the board of Cardiff City and Tan Sri Vincent Tan for giving me the opportunity. The club's owner Mr Vincent Tan invested heavily in the club and supported our decisions in our push for promotion to the Premier League. Without him this would not have been possible.
"My thanks to all those at the club and the fans who have supported me during my time in Wales. If I have caused any offence to any one during this time, especially to Tan Sri Vincent Tan, then I apologise without reservation.
"I wish everyone associated with Cardiff City FC all the best for the future and thank them for their support."
Following Mackay's departure, assistant David Kerslake was briefly in charge before Solskjaer presided over the club's relegation.
Cardiff said the resolution of the matter would help the club move forward ahead of their return to the Championship.
"The agreement represents a positive outcome for the club," they said via their official website.
"We now look forward and focus fully on planning for next season and the challenge of regaining Premier League status."
