Messi was sent off, along with Gary Medel, after an altercation in the first half of a tempestuous meeting at Arena Corinthians.
After the match, which Argentina won 2-1 thanks to first-half goals from Sergio Aguero and Paulo Dybala before Arturo Vidal scored a penalty, Messi did not appear with his team-mates to collect his medal.
When asked why he was not on the pitch for the ceremony, he said: "Because we don't have to be part of this corruption, this lack of respect towards the whole Copa."
Messi was dismissed when Medel reacted angrily to a challenge and repeatedly bumped into the Barcelona star with his chest. Neither Medel nor Messi raised their hands and the Chile man was the clear aggressor in the incident.
🔴🔴 MESSI OFF!! #LionelMessi SENT OFF with @MedelPitbull for probably the weirdest fight in football history 🥊😳 | 🇦🇷 2-0 🇨🇱 | Watch LIVE on beIN SPORTS CONNECT ▶️ https://t.co/vL0COHZ8AD | #CopaAmerica | #ARGCHI pic.twitter.com/sb5V4ukV9B— beIN SPORTS (@beINSPORTS_AUS) July 6, 2019
There was a VAR (video assistant referee) check but referee Mario Diaz de Vivar did not look back at the footage on the pitch-side monitor.
Messi blasted the officiating at the tournament after Argentina lost to host Brazil in the semi-finals in a match in which Lionel Scaloni's side had two penalty appeals that went unchecked by VAR.
He suggested the red card had been punishment for those comments, saying: "With a yellow card it would have been over for both of us, but, well, maybe I'm paying for what I said last time.
"The important thing is that the team finished the tournament well. We do not have get caught up in the lack of respect that happened here.
"I leave calmly, with my head high and proud of this group, who gave everything. I hope that this group is respected — it has a lot to give.
"The group showed a lot of personality. It's not easy to start with a defeat [against Colombia] and then drawing [against Paraguay]. That made us stronger and we finished in a very positive way.
"They could have reviewed the sending off, but VAR was not used."
Looking ahead to the final between the Tite's men and Peru, Messi said: "Hopefully, the referees and VAR won't influence it and they'll let Peru compete, but that looks difficult to me."
He added: "I always tell the truth. That leaves me calm. Then, if what I say has repercussions, it is not to do with me as I am always sincere."
Messi continued to complain via social media after the match, claiming football hadn't been fair to his team, which he believes was "superior" to Brazil and should have been in the final.
"We leave this Copa with a triumph but above all with our heads up and with the feeling that this time football was not fair with us," Messi wrote on his Instagram account.
"For what we played and because we were superior to Brazil, we deserved to be in the final tomorrow.
"But we have to look forward and with optimism because there is a future and a very large base in this team and we just have to give time."