The Uruguayan forward was given a 10-game ban for violent conduct following the incident during the second half of the sides' 2-2 draw on April 22.
"Luis Suarez owes Liverpool big time and he shouldn't be thinking about leaving this summer," Aldridge wrote in his Liverpool Echo column.
"The striker needs to go to anger management classes, get himself sorted and then come back next season and show everyone how good he is.
"Winning Player of the Year and the Golden Boot next season would be the perfect response to those people in authority who seem intent on forcing him out of English football."
Liverpool did not appeal against the decision to ban Suarez, meaning that the former Ajax man will be out of action in the Premier League until late September.