After Kohli said on the eve of the third Test that he had no regrets over accusing Australia of systematically abusing the decision review system (DRS) in their last match in Bangalore, Smith angrily denied that he was guilty of more than a momentary lapse.
While the International Cricket Council is set to hold clear-the-air talks between the two men, the new exchanges are likely to complicate their efforts to broker peace between the top two-ranked Test teams.
Kohli's accusations last week had plunged the teams' often tense ties back into crisis mode, with Australia's board calling his comments "outrageous", prompting an equally robust response from the Indian camp.
Speaking to reporters ahead of the third Test in Ranchi, Kohli said his focus was now on the cricket but he refused to back down on his claims against Smith.
"I don't regret anything that I have said but at the same time it's very important not to be stupid and go on about the same thing on a daily basis," said Kohli.
India were furious when Smith was spotted looking up at his dressing-room for guidance in Bangalore on whether to seek a review of the umpire's decision after being given out lbw -- a clear breach of the DRS rules.
While Smith called the incident on the fourth and final day a one-off, Kohli said after the match that it had been going on for days.
Asked on Wednesday about Kohli's latest comments, Smith said he was keen to move things on but still snapped back at his counterpart.
"Virat obviously stuck by his comments. From my point of view they are completely wrong. I obviously came out of the game and said I made a mistake," Smith told reporters.
"In regards that we do it consistently is absolute rubbish in my opinion. I think he was wrong in his statement."
While Smith did reveal that he had met match referee Richie Richardson, there has so far been no clear-the-air meeting between the two skippers. The four-match series is tantalisingly poised at 1-1.
"I just met with Richie ... and the other umpires. Basically what they said was 'we want to see cricket as the winner in this Test match (and) both teams to play within the rules of the game'," said Smith.
"And they said, 'we know both teams, Australia and India Test matches are fiercely contested - number one and number two in the world. It's just about playing within the boundaries of the game'."
"Both Virat and I are meeting with Richie ahead of the game tomorrow. I think it will be a similar sort of thing with Richie ensuring that we are both here to play the game of cricket," he added.
India and Australia, the top two-ranked teams in Test cricket, have often been at loggerheads and relations hit a low point in 2008 when Indian spinner Harbhajan Singh was accused of calling Andrew Symonds a monkey.
Kohli himself was fined 50 percent of his match fee three years ago for bringing the game into disrepute after flashing the middle finger to the crowd in the Sydney Test.
The Indian skipper said he was happy to put the events of the last Test behind him, admitting too much bad blood has flowed in the past.
"I think it was a mature decision on everyone's part to move on. We have seen instances in the past stretch on too long -- it just causes disharmony and there is no outcome to it," he said.
"It's time we focus on the remainder of the series. There's a lot of cricket to be played and it should not happen in bad taste."
After the match in Ranchi, the series reaches its climax in Dharamsala in a match starting on March 25.