Barcelona forward Messi, 32, was sent off in Argentina's 2-1 third-place play-off win over Chile and later said the "cup was fixed for Brazil".
He has also been fined $50,000 (£41,121) by Conmebol, the South American Football Confederation.
Messi has seven days to appeal Conmebol's decision to suspend him.
The ban means Messi will miss Argentina's upcoming friendlies against Chile, Mexico and Germany in September and October.
Argentina's qualifying campaign for the 2022 World Cup begins in March 2020.
Following Argentina's 2-0 semi-final defeat by hosts Brazil, the Argentine Football Association complained about "serious and gross refereeing errors".
In response, Conmebol said accusations questioning the integrity of the Copa America were "unfounded" and "represent a lack of respect".
Messi was shown a red card in the 37th minute against Chile, following a clash with Gary Medel, who was also dismissed.
"We don't have to be part of this corruption," Messi said. "They have showed us a lack of respect throughout this tournament.
"Sadly, the corruption, the referees, they don't allow people to enjoy football, they ruined it a bit."
BBC Sport
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