Nel appeared before a SANZAR disciplinary committee early on Sunday morning where he received the punishment, meaning he will now fly back to South Africa and take no further part in the Bulls' Super 14 tour of Australasia.
A SANZAR statement read: "Nel was cited to appear before Judicial Officer Terry Willis on a charge of striking Waratahs player Rocky Elsom in the March 10 game between the Waratahs and the Bulls at Aussie Stadium.
"The charge came out of an incident in the 70th minute. After hearing evidence from JP Nel and submissions from his representative Ian Schwartz, and bearing in mind the player's plea of guilty, previous good record and obvious remorse, the Judicial Officer found the appropriate sanction to be three weeks."
Nel, however, was unhappy at the outcome.
"I'm extremely disappointed with the ban and have been shattered by the length of it," he said.
"I'm going back to South Africa but it is disappointing as something special is happening in this Bulls team at the moment and I would have wanted to make a contribution right until the end of the tour.
"I know I can't do anything about it now but will work hard to ensure I am still in top condition when the team returns home for their final four games."
The Bulls have won two games in Australia on the road after previously never having recorded a victory in Canberra or Sydney in 12 years of competition.
They now face defending champion Crusaders in Christchurch on Saturday before playing the Highlanders and Hurricanes.