The 42-year-old Scot said he was not racist, sexist, homophobic or anti-Semitic but accepted his behaviour had been "completely unacceptable".
Mackay has come in for a wave of criticism after his former club Cardiff submitted a dossier to the Football Association regarding his conduct and that of ex-head of recruitment Iain Moody.
Mackay engaged in sending what have been reported as discriminatory, mocking texts, with the controversy apparently costing him the chance to get back into management with Crystal Palace, who were understood to be readying his appointment but are now looking elsewhere for their next boss.
He admits sending key texts at the centre of the outrage at a time when his position was under scrutiny at Cardiff, who sacked Mackay last December despite him guiding the club into the Premier League for the first time in their history.
Mackay said: "Out of 10,000 text messages in and out of someone's phone I sent three and that being the case, looking at them they are completely unacceptable, inappropriate and for that and for any offence I've caused I sincerely apologise for that.
"I've received some but the three that I've sent are the ones that I'm accountable for and I feel I shouldn't have sent them.
"I did it in a period where 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.
"But that doesn't excuse anything and was unacceptable."
He added: "I'm a manager, I'm a leader of people and it should not have happened. But before all that and foremost, I'm a human being and I made a mistake.
"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 am not anti-Semitic."
He added: "These are testing times - make no mistake about that. But I've got values and resilience and I've got a love for British football and I will come back from this."
Mackay is seeking help following his conduct, saying: "It's certainly something I've been involved with, speaking to my union, asking them about equality and diversity training and it's something I'll be going forward with."
The fallout from the scandal has been colossal, with the League Managers' Association coming under fire for defending Mackay, insisting in a statement on Thursday night that the texts were merely "friendly banter".
Cardiff called for LMA chief executive Richard Bevan to step down from his role.
Cardiff stated: "We.. find it entirely reprehensible that the LMA should itself put out a statement which seeks to dismiss deeply offensive racist comments as 'friendly banter'. If that is the view held by the LMA, as appears from its statement, we consider that Richard Bevan's position is untenable and we call for his resignation."
Mackay admitted the term "banter" was inappropriate as a defence, but said the statement was otherwise "very, very accurate".
When asked if he thought Bevan should resign, Mackay said: "No I don't, I absolutely don't."
Mackay added: "I've certainly never used the word 'banter' and that is not something I personally feel that it was, but I think the overall statement was very, very accurate.
"I agree that word is wrong and the LMA soon realised that and they've tried to rectify that today."
In its apology, released prior to Cardiff's statement, the LMA said: "The LMA apologises for some of its wording, in its release yesterday, which was inappropriate and has been perceived to trivialise matters of a racist, sexist or homophobic nature. That was certainly not our intention."
That clearly did not pacify Cardiff, who also said the LMA had known about the existence of the offensive messages involving Moody and Mackay for months and had therefore been "complicit in the attempt to conceal these messages".
Cardiff's statement added: "When the messages came to light, over three months ago, the club strongly encouraged and advised Mr Moody and Mr Mackay to deal with the issue directly with the FA.
"It was made clear to them, and their LMA-appointed lawyers, that the nature of the communications meant the club was under a duty to report their findings to the FA if they did not take appropriate action themselves.
"Nevertheless, with the backing of the LMA, Mr Moody and Mr Mackay chose to do nothing. The LMA were therefore complicit in the attempt to conceal these messages [of which there were many more than the two isolated texts acknowledged by Mr Mackay]."