Kosar criticised Rams quarterback Kellen Clemens, the receivers and receivers coach Ray Sherman during the match, although he has since called up coach Jeff Fisher to personally apologise for his comments.
Browns CEO Joe Banner branded Kosar's behaviour "personal and unprofessional", but insists that he now considers the situation resolved.
"We don't condone the personal and unprofessional approach that Bernie took with some of his comments during the broadcast Thursday night," wrote Banner in a team statement.
"We've spoken to Bernie, he understands that, and we would expect the situation is resolved moving forward. We've also reached out to the Rams organization and have shared those same sentiments."
Kosar spent eight season with the Browns between 1985 and 1993.