Both players will be on the carpet for "conduct that brings the game into disrepute".
According to media reports, the incident has been handed in the hands of on-field umpires and match referee.
Earlier on Tuesday, Australian coach Darren Lehmann said that he was "certainly supporting" Warner, and that his position as the vice-captain was not in jeopardy.
The war of words between Australia and South Africa - itself a product of a war of words between Warner and de Kock - has escalated quickly.
Crowe is also expected to meet with skippers Steve Smith and du Plessis at some point this week, imploring them to take control of their teams and ensure the next match is played in better spirits. "We want to play a way that gets us success and we've just got to make sure we don't cross the line - that's the key", Lehmann said.
Their teams have until Wednesday to respond but, while de Kock can only receive a maximum of two demerit points, Warner's level-two charge means he stands to be hit with three or four points.
With both sides blaming the other for sparking the incident by making personal insults, Lehmann said: "Everyone has their own opinion".
Taking on to twitter, Warne wrote, "Chat, banter, sledging has always been a part of any series between SA & Oz".
Shane Warne on David Warner spat: Stop the whingeing!
"The International Cricket Council today announced that Australia vice-captain David Warner and South Africa's Quinton de Kock have been reported for breaching the ICC code of conduct following their altercation on the fourth day of the Durban Test which was captured on CCTV", the ICC said in a statement.
Former England captain Michael Vaughan suggested Warner had a reputation as a sledger and it was only a matter of time before insults were hurled back.
A claim later emerged on Monday night that de Kock's personal slur directed at Warner was in response to Warner first crossing the line with a series of personal attacks directed at the South African keeper as he was trying to save the game with his bat in the second innings.
Smith's version of events was disputed by South African manager Mohammed Moosajee, who blamed Warner.
"I'm sure the captains will chat and the coaches will chat and we'll get away to play the game", he added. If you are saying something you've got to take it and that's the opinion of Quinton.
"Respect is the key & and I hope nothing personal was said to any player towards anyone from either side".
"We play our best cricket when we're aggressive, we're in the fight together, we're hunting as a pack", Smith said.
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