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Wicketkeeper Paine at one stage called Ashwin a "dickhead" as the spinner and Hanuma Vihari batted through the last three hours of the match to save a draw at Sydney Cricket Ground and keep the series tied at 1-1.
Paine, who apologised to his team mates for his three dropped catches in his post-match news conference on Monday, requested the opportunity to address the media again on Tuesday.
"I want to apologise for the way I went about things, I'm someone who prides himself on the way he leads this team and yesterday was a poor reflection of that," he told reporters.
"My leadership wasn't good enough, I let the pressure of the game get to me. Yesterday, I fell short of my expectations and my team's standards.
"I'm human, I want to apologise for the mistakes I made yesterday ... we've set really high standards over the last 18 months and yesterday was a bit of a blip on the radar."
The 36-year-old said he had felt "off" for the entire match, during which he was also fined 15% of his match fee for swearing while complaining to an umpire.
Paine is acutely aware that the only reason he was handed the Australian captaincy after the Newlands ball-tampering scandal cost Steve Smith the job in 2018 was because of the non-confrontational way he went about the game.
"I spoke to (Ashwin) quickly after the game yesterday, and I said to him 'yeah, you end up looking the fool, you open your mouth and you end up dropping a catch'," he added.
"We had a bit of a laugh about that ... and everything was fine."
The wicketkeeper also defended Smith against charges of gamesmanship during the Sydney test.
He said Smith was not erasing Rishabh Pant's guard in the batting crease but merely indulging one of his many idiosyncrasies.
Video footage from the stump camera of Smith dragging his boot across the crease on a tense final day of the match went viral on social media and some former players accused him of gamesmanship at best, cheating at worst.
Paine said Smith had been upset by the allegations.
"I know he's really disappointed with the way it's come across," Paine told reporters on Tuesday.
"If you've watched Steve Smith play test cricket, that's something that he does every single game five or six times a day.
"He's always standing in the batting crease shadow batting. As we know, he's got those many Steve Smith quirks and one of those is he's always marking centre.
"He was certainly not changing (Pant's) guard and if he was, that's something that the Indians would have kicked up a bit stink about at the time."
Paine said his own behaviour had not caused him to consider his position as captain and he was looking forward to leading Australia out for the final test of the series at the Gabba on Friday.
"I have to cop what's coming on the chin and I'm really looking forward to moving on to Brisbane and getting back more for playing the game more like Tim Paine plays the game," he said.