The exchange followed the Spurs' 97-86 victory in Sacramento on Friday night. The volatile Cousins will lose about $70,500 of his $3.9 million salary.
Kings coach Keith Smart has discussed the incident with Cousins.
"Any little mark is gonna knock you all the way back down again," he said, remembering their conversation. "So you've got to stay one step ahead and be perfect, with everything that goes on. You've got to be a man now and accept what's happening. And then we've got to move on. But understand that everything you do until you are this person that we believe you can become, you have to do things perfect, because when you make mistakes they affect not just you but it affects our entire team.
"So let's put this behind you, we've got the right people in place now to get you through all these things and we're looking forward to the polished person to eventually come out."
Cousins will sit out games at the Los Angeles Lakers on Sunday and at home against Portland on Tuesday.
Elliott was a longtime Spurs player who is the team's primary TV analyst.
Information from ESPN.com's Justin Verrier and The Associated Press was used in this report.