The International Automobile Federation (FIA) said in a statement on Tuesday that its president Jean Todt had decided, "on an exceptional basis", not to take disciplinary action against the four-times world champion.
The FIA made clear, however, that any such incident in the future would be met with a summons to its International Tribunal.
Vettel could have faced anything from a reprimand to a race suspension if the governing body had decided to take disciplinary action.
The FIA said the German "spontaneously sought out" Whiting after the race and had, on his own initiative, also sent letters to him and Todt "in which he apologised profusely for his actions".
"He also indicated that he would likewise be contacting (Red Bull's) Max Verstappen and vowed that such an incident would never occur again."
Vettel's outburst, in the closing laps, was broadcast to a worldwide television audience on Sunday with the swear words beeped out.
Whiting drew his ire after race direction did not immediately order Verstappen to let Vettel past when the Dutch teenager gained an advantage by cutting a corner while defending third place.
In a farcical conclusion, Vettel was promoted to third after the race and appeared on the podium before being demoted back to fifth for an illegal defensive move of his own against Verstappen's team mate Daniel Ricciardo.
The Australian was moved up to third and Verstappen, who had finished third on the road but dropped to fifth after a five second time penalty, fourth.
Red Bull team principal Christian Horner said on Sunday that what Vettel had done was similar to a football player abusing the referee and he expected some sort of punishment for his former driver.
Whiting, who is also the FIA's technical head, is a highly respected figure in the paddock.
"The FIA will always condemn the use of offensive language in motor sport, especially when directed at officials and/or fellow participants," the governing body said.
"(It) expects all participants in its Championships to be respectful and mindful of the example they set for the public and the younger generation in particular."
Vettel is enduring a tough time at Ferrari, failing to win a race so far this season with only two of 21 rounds remaining.
(Editing by Ed Osmond and Toby Davis)
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