In a statement, CONMEBOL dismissed the star's comments as unacceptable and unfounded.
After Brazil beat Peru 3-1 to win its ninth South American title, Casemiro tried not to be drawn on Messi's comments, but offered a brief response.
"Those who have a mouth speak what they want," the midfielder said. "It's not my turn to speak. It's a delicate issue."
Goals from Everton and Gabriel Jesus and a Richarlison penalty saw Brazil past Peru despite a second-half red card for the Manchester City forward.
The officiating has drawn plenty of criticism at the Copa America, but Casemiro was unwilling to judge the referees' performances.
"It's not for me to say if the referee did well," he said. "We all try to do the best and we must congratulate Peru for the great Copa they did."