"Those sorts of things are not on from both sides," Smith said at his post-match press conference after Australia won the first match in the four-Test series by 118 runs, completing the triumph in just 20 minutes on Monday's final day.
"Getting personal on the field is not on. We were certainly very chirpy out in the field as well. As far as I'm aware, we didn't get personal towards Quinton," Smith added.
"What he said got a little bit personal towards Davey and, as we saw, it certainly provoked an emotional response."
CCTV footage released overnight showed Warner being restrained by team mates Usman Khawaja and Nathan Lyon as he directed a verbal barrage at De Kock when the players were heading back to the dressing rooms at tea on Sunday.
At the time, South Africa were fighting their way back into the contest after losing their top order cheaply in the pursuit of a mammoth 417-run victory target.
Australia wicketkeeper Tim Paine also moved in to block Warner before captain Smith ushered the opening batsman away.
"It's rather unfortunate that the incident took place and certainly not in the spirit of the game," South Africa's team manager Mohammed Moosajee had said earlier in a statement.
"The match referee had spoken to us after the day's play. We will await any further communication from the ICC around this incident."
Cricket Australia (CA) said it was also looking into the incident.
"CA is aware of reports of an incident between players in Durban," a spokesperson for the Australian governing body said.
"CA is working to establish the facts of what has occurred and will not be commenting further until that has been done."
A fired-up Warner had earlier directed a verbal volley at Proteas opener Aiden Markram after the rookie batsman was culpable in the running out of AB de Villiers for a duck.
Former South Africa captain Graeme Smith criticised opening batsman Warner as "a bit of a fool" and "like a fly buzzing around your head" but Markram, who went on to score an impressive 143, played down the sledging as part of the game.
De Kock, who was the last batsman out for 83 on Monday, was seen shaking hands with Warner after the match had ended.
The International Cricket Council (ICC) have yet to comment on the issue or a charge against Australia spinner Lyon for dropping the ball onto a prone De Villiers after he was run out before lunch on Sunday.
AB de Villiersaiden markramcricket australiaDavid WarnericcKingsmead TestNathan LyonQuinton de Kocksouth africa vs australia 2018steve smith
First Published: March 6, 2018, 7:30 AM IST