Smith, ahead of what will be his first series since taking over from Michael Clarke, was reacting to criticism by McCullum over an incident during Australia's tour of England this year.
Smith's failure to withdraw an appeal after England player Ben Stokes was given out obstructing the stumps from an attempted run out while taking evasive action, drew a sharp opinion from McCullum.
"We've all done things on the field that we regret later, I know I certainly have," McCullum wrote in a column in Britain's Daily Mail newspaper.
"But it was disappointing that Smith had a chance to make a statement about the way he wants his side to play the game and chose to go the other way.
" ... By not withdrawing the appeal, Smith showed his immaturity. He may live to regret it."
Smith admitted he was "a little bit disappointed" by McCullum's comments.
"I didn't really think it was any of his business," he told Sydney's Daily Telegraph.
He insisted he had moved on from the incident but "wouldn't change a thing" if the incident was replayed. "No regrets," he added.
While Australia's hefty sledging of New Zealand batsmen during the World Cup final left a sour taste, Smith won't shy away from being tough-minded in their on-field approach.
"I think that's the way we play our best cricket - if we're aggressive and positive," he said.
"We know there's a line that can't be crossed and our players are going to be playing that hard aggressive cricket and making sure we stay on the right side of that line."
New Zealand had a solid training session at Brisbane's Allan Border Field on Sunday, and another workout at the Gabba nets yesterday, compensating for the shambles of a leadup match in Sydney last week, when the game was abandoned halfway through the second day. They are in good spirits.
"As a kid growing up and watching these guys play and how they do it, to get a chance here is pretty special," offspinner Mark Craig said. "I know the rest of the boys are really excited for the series to get under way."
Craig is expecting bounce and carry, which are the hallmarks of Gabba test pitches, and he's hoping with two left armers in the Australian side, Mitchell Starc and Mitchell Johnson, plus teammate Trent Boult, there will be some footmarks for him to work with as the test wears on.
