The South American football confederation said questioning the integrity of the Copa America is "unacceptable" after Lionel Messi alleged Argentina were victims of "corruption" in their 2-1 win over Chile.
Messi saw red towards the end of the first half of their third-place play-off in Sao Paolo and refused to take part in the medal ceremony.
The Barcelona star was dismissed following a confrontation with former Cardiff midfielder Gary Medel, who was also shown a red card.
Messi appeared to have been hard done by, with Chile defender Medel appearing to push his head towards the Argentina playmaker as they squared up, chest to chest.
The Argentina captain suggested he had been singled out following his comments over the "crazy" officiating in the wake of their 2-0 semi-final defeat by hosts Brazil in Belo Horizonte.
"We don't need to be part of the corruption that we've suffered at this tournament," Messi said, as quoted by Spanish media outlet AS.
"Medel is always right at the limit. With a yellow that would have been the end of it for both of us, but well, maybe what I said recently had an impact.
"What's important is that the team finished well in the tournament. Maybe this was ordered and I ended up suffering because of what I said."
In the aftermath, CONMEBOL said a "fundamental pillar of fair play is to accept the results with loyalty and respect".
Although it does not mention Messi by name, the statement said: "The same goes for refereeing decisions, which are human and will always be improvable.
"It is unacceptable that as a result of incidents typical in competitions, involving 12 teams, all on equal terms, unfounded accusations have been launched that lack the truth and question the integrity of the Copa America.
"These accusations represent a lack of respect for the competition, all the participating players and the hundreds of professionals of CONMEBOL, an institution that since 2016 has been working tirelessly to make transparent, professionalise and develop South American football."
Messi had helped set up an early goal for Manchester City forward Sergio Aguero, with Argentina going 2-0 up through Paulo Dybala.
Chile were handed a way back into the match following another controversial VAR review which resulted in a penalty being awarded after Argentina's Giovani Lo Celso was adjudged to have fouled Charles Aranguiz. Arturo Vidal dispatched the spot-kick.