The French striker put a string of angry posts on Twitter and Instagram on Saturday aimed at those who had criticized him for eating a gold-coated steak, AFP reported.
"I had a long talk with Franck and I let him know that we would impose a heavy fine on him and he accepted it," Salihamidzic told journalists in Qatar, where the club is on a training camp.
"He used words that we, FC Bayern, cannot accept and that Franck does not have the right to use, as a role model and player of FC Bayern", added Salihamidzic.
On Thursday, the 35-year-old Bayern Munich player posted a video on social media showing him in a steak house owned by celebrity Turkish restauranteur Salt Bae and rubbing his hands before tucking into a huge chop coated in gold.
That ostentatious luxury, and the reported cost of the meal, drew ferocious criticism before Ribery – who earns an estimated €8 million a year with the German champion – hit back on Saturday.
Salihamidzic made clear that Ribery did not pay the bill himself because he had been invited to the restaurant. He also acknowledged that the player had been responding to a flurry of insults.
"Franck has been slandered and insulted," said Salihamidzic. "And not only Franck, but his pregnant wife, his child and his mother, who is currently in hospital for an operation. Franck wanted to defend himself and defend his family. He has the right to do that. I support him in that, but unfortunately he totally lost control."
German media and social media users had called on the reigning Bundesliga champion to punish the striker, who is out of contract at the end of the season.
"Bosses must suspend Ribery," urged German tabloid Bild.
The paper referred to comments by Karl-Heinz Rummenigge, the club's chief executive, who, in October responded to media criticism by quoting the German Constitution that "the dignity of the human being is intangible".
"He is right, but it would not hurt if he explained it to his employee Franck Ribery," said Bild.
Bayern and Rummenigge have also been on the defensive recently after criticism from some of the club's own fans over the decision to return to Qatar for winter training.