Reports emerged on Thursday that owner Vincent Tan had offered an ultimatum to the Scot - resign or be sacked.
Luis Suarez heaped misery on the Welsh club with two goals and an assist for Raheem Sterling at Anfield on Saturday, but Mackay was defiant.
"I reiterate 100 per cent that I won't be resigning from the football club," he told BT Sport.
"I'm a proud and passionate man to lead this football club, to lead this team. I have done for two and a half years and I won't be leaving."
Questioned as to why his position had reportedly come under threat from Tan despite guiding the club to the Premier League, the 41-year-old said: "That is something you will have to ask other people."
Jordon Mutch reduced the deficit for the visitors in a greatly improved second-half display for Cardiff and Mackay was proud of his side's efforts as well as the support given to them by the travelling fans, who stayed long after full-time to show their backing for the manager.
"The crowd were our 12th man," Mackay added.
"You see the fans, how much they're backing the team, showing how much we should be enjoying the Premier League this season. They've been like that all season.
"I'm proud I represent them; that my staff and my group of players represent them and I can tell you we're absolutely going to make a fist of this."
