Mackay, who left Cardiff under a cloud last year following a falling-out with club owner Vincent Tan, apologised on Friday after English media published details of texts he sent while in charge at the Welsh club, who were relegated from the Premier League last season.
"I've been in a multi-cultural football environment for 20 years. I love British football and I am no racist. I am no sexist. I am no homophobe and I'm not anti-Semitic," he said in a TV interview.
"I sincerely apologise. It was something I did, and there's no excuse for that. It was in a period (when) I was under immense pressure and stress in terms of the relationships that were possibly not going too well at my football club at the time."
According to the Daily Mail, which broke the story, the text messages under scrutiny were exchanged between Mackay and Cardiff's then sporting director, Iain Moody - who resigned from a similar role at Crystal Palace on Thursday.
Mackay, 42, said he was responsible for three offensive messages.
"Obviously, it was someone else's phone that a vast array, over 10,000, of private texts were lifted from.
"I received some (of these texts) but as I say it's the three texts I'm accountable for and that I shouldn't have sent."
Former Scotland defender Mackay was sacked by Cardiff midway through last season after a public disagreement over transfer spending.
He had been expected to take charge of top-flight Crystal Palace this week but was ruled out of the running after details of the messages emerged.
Mackay also defended his dealings in the transfer market as Cardiff manager and said he was not guilty of any wrongdoing.
"We were a small Premier League club and you have to compete with the bigger clubs for the better players and we brought in some good players to the club," he said.
"We tried to do the best with our recruitment. No one ever gets it 100 percent right. But people need time, especially if they're injured," he added.
"In the five months I was there we managed to stay out of the bottom three with that group of players."
Asked if he had any regrets about the players he had brought to the club, he said: "None of them.
"The signings were done in a fashion where there was a process that had been long-standing. I signed 31 players in two years for the football club with the process in place, with various people signing off and looking at it," he said.
"The players we brought in were scouted and looked at. Some take longer to settle, some don't perform as well as you think. Some hit the ground running right away. Others go to World Cups.
He also defended Moody's role in the transfer deals.
"You can never be 100 percent certain on someone else but Iain Moody is someone that worked hard for Cardiff City football club, and Watford before that, and at every point, as far as I'm aware, did things in the best interests of the football club."