Messi was controversially sent off in the first half as Argentina defeated Chile 2-1 in Saturday's third-place play-off at Arena Corinthians.
Argentina, who lost 2-0 to hosts Brazil in the semi-final, received the bronze medal but Messi did not appear with his team-mates during the presentation as the captain railed against "corruption" and "a lack of respect".
Messi continued to be vocal post-match, however, the five-time Ballon d'Or winner believes the future is bright for Argentina.
"We leave this Cup with a triumph but above all with our heads up and with the feeling that this time football was not fair with us," Messi said in an Instagram post.
"For what we played and because we were superior to Brazil, we deserved to be in the final tomorrow.
Nos vamos de esta Copa con un triunfo pero sobre todo con la cabeza alta y con la sensación de que esta vez el fútbol no fue justo con nosotros. Por cómo jugamos y porque fuimos superiores a Brasil, merecíamos estar en la final de mañana. Pero hay que mirar hacia adelante con optimismo porque hay futuro y una base muy grande en esta Selección y solo hay que darle algo de tiempo. #VamosArgentina 🇦🇷
"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."
Messi was dismissed when Chile's Gary 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 as both players saw red.
There was a VAR 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 Copa America after Argentina were eliminated by Brazil in the final four 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."
With Brazil set to face Peru in Sunday's decider, Messi added: "Hopefully, the referees and VAR won't influence it and they'll let Peru compete, but that looks difficult to me."
Messi said: "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."