The veteran Ivory Coast forward posted a now-deleted video on Facebook that showed him shaking hands with team-mates and a member of staff.
Kalou bumped fists before speaking to Vedad Ibisevic, with the Hertha captain seemingly complaining about the pay cut he took as part of the response to the ongoing coronavirus heath crisis.
The ex-Chelsea player also interrupted what appeared to be a COVID-19 test being undertaken by defender Jordan Torunarigha.
A statement from Hertha read: "With his video, Salomon Kalou gave the impression that Hertha BSC's players are not taking the required social distancing and hygiene guidelines of health services seriously.
"Hertha BCS would like to clarify that these were the actions of one individual player. The fact that other team members did not draw his attention to this transgression, and instead returned the greeting with a handshake, makes it clear that rules on social distancing must be enforced with even greater intensity.
"With this video taken inside the team's dressing room, Kalou broke clear internal rules and displayed a behavior that is neither appropriate for this current situation nor reflective of the code of conduct of Hertha BSC. The club has therefore made the decision to suspend the player in question from training and matches with immediate effect.
"The discussion on partially waiving the salaries of the Hertha BSC's professional players was caused by incorrect calculations. These have since been corrected.
"These measures were openly discussed in coordination with everyone and supported by all of the players. There was no internal criticism. On the contrary, everybody welcomed these measures."
Kalou apologized in the statement, saying: "I'm sorry if I gave the impression that I'm not taking coronavirus seriously. I want to apologize for this.
"The reality is the exact opposite, because I'm really worried about the people of Africa, because health systems aren't as good there as they are in Germany.
"I didn't really think it through and was excited that my tests came back negative. I would also like to apologize to all those who appeared in the video, who didn't know that I was broadcasting live and whom I didn't want to bring into this situation."
Having detailed such steps as part of its attempts to secure a restart to competitive action – perhaps as early as the middle of this month - the league tweeted a short statement strongly criticising Kalou.
"The pictures of Salomon Kalou from the dressing room of Hertha BSC are absolutely unacceptable," the statement said.
"This cannot be tolerated when other players and clubs have stuck to the guidelines because they have grasped the seriousness of the situation."