Argentina beat Chile 2-1 on Saturday to finish third in Brazil, but Messi didn't attend the medal ceremony after he was shown the second red card of his career before half-time.
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"In football sometimes you win and sometimes you lose and accepting the outcome of a match with loyalty and respect is a fundamental pillar of fair play. Same goes to the refereeing decisions, that are human and will never be perfect.
"It's unacceptable that because of incidents proper to the competitions, where 12 nations competed in, all in equal condition, there have been unfounded accusations that misrepresent the truth and put to judgement the integrity of Copa America."
The statement concluded: "Said accusations represent a lack of respect for the competition, all the football players involved and the hundreds of professionals of CONMEBOL, an institution that since 2016 has been tirelessly working to professionalise and develop South American football."
Messi, 32, and Gary Medel got into a physical exchange in the 36th minute, which led to both players being dismissed by referee Mario Diaz de Vivar in one of several controversial incidents, via Sky Sports (UK only):
Argentina were knocked out of the tournament semi-finals following a 2-0 defeat to Brazil on Tuesday, and Messi was clear in his belief that he felt there was wrongdoing at play.
The Barcelona talisman is usually regarded as well tempered, but his frustrations carried on off the pitch as Messi told reporters he felt the Copa was "set up for Brazil" and against his own team:
"There is no doubt, the whole thing is set up for Brazil. I hope the VAR and referees don't play any part in the final and that Peru can compete but it looks tough to me.
"I did not want to be part of this corruption, we shouldn't have to be part of this disrespect we suffered during the Copa America. We could have gone further but we weren't allowed into the final. Corruption, referees and all the rest stopped people from enjoying the football.
"I always tell the truth and I'm honest, that is what keeps me calm, if what I say has repercussions that is not my business. I think that what happened is because of what I said [against Brazil]."
The only other red card Messi has received during his 15-year professional career came in his Argentina debut against Hungary:
Pisani noted referee Diaz de Vivar could have used the pitch-side monitor to follow up the video assistant referee's initial check on the red card fracas but opted not to.
Even Medel—despite their earlier clash—agreed with Messi and said their dismissals could have been handled better, per broadcaster Juan Arango:
Juan G. Arango 🇨🇴 🇮🇹 @JuanG_Arango
Medel: "I agree with Messi. A yellow sufficed. It was in the heat of the moment. The referee was bad. He ruined everything that was good about Argentina-Chile. Argentina were harmed in the match against Brazil. We were sent to a hotel far away prior to the Colombia match." https://t.co/e4lcuX2uVA
Brazil are unbeaten in the tournament and are yet to concede a goal in their five Copa outings. But they too have looked uncomfortable at times and have come out on the losing side of some VAR incidents, namely during their 0-0 draw against Venezuela in the group stage.
Tite's side entered their home tournament on the back of a 10-game unbeaten run and are without injured star Neymar, while Argentina have severely lacked consistency in recent times.
The Selecao deserve their place in the final based on recent performances, and they'll hope to replicate their 5-0 group-stage win over Peru when the two clash on Sunday.