Tan hit out at Mackay via a statement from club chief executive Simon Lim, who said the owner was "extremely upset" that the manager had gone public with his desire to strengthen his squad in January.
It is the latest setback in the relationship between Tan and Mackay, who has guided his side to 15th place in the Barclays Premier League after Saturday's 1-0 win over West Bromwich Albion.
In an interview broadcast on Tuesday on BBC Radio Wales, Mackay said: "I will not resign from my post. I was surprised obviously, but at the same time my job doesn't change.
"It's a job only three-quarters done as far as I'm concerned - making Cardiff a sustainable Premier League team.
"I have a great group of players who are competing at this level and enjoying this level. The fans are enjoying being at this level and my duty is to make sure we continue to do that.
"It would be wrong of me to walk away from a job because the owner doesn't want to give me more money in January."
Mackay had infuriated Tan with comments made after the win over the Baggies, in which Mackay specifically stated he was looking to make three new additions to his squad during the transfer window.
Fans expressed dismay at the latest off-field distraction with the Cardiff City Supporters Trust once again giving their full backing for Mackay and calling for Tan to restore unity within the club.
Trust press officer Phillip Nifield told the Press Association: "We've got enough problems staying in the Premier League without all the off-field stuff which is really unnecessary, and really disappointing.
"It's perfectly normal for a manager to draw up a wish-list before the January transfer window and you would be surprised if he didn't want extra players. The chairman's surprise is a surprise in itself."
Nifield added that he feared for Mackay's long-term future at the club, saying: "There is usually only one winner between an owner and a manager but we don't blame Malky for not wanting to walk away.
"Malky has achieved something we never had in terms of taking us to the Premier League and the fans in general would be disappointed if he went because of what has happened with Mr Tan.
"The owner has scored a few own goals in terms of changing the colours of the badge, and while the fans generally appreciate the financial clout he has brought us, it should not mean we are not allowed to question the sort of things he's done."
Source: PA
Source: PA