CONMEBOL announced the decision with a statement on Friday, asserting Messi's claims "were unfounded" and "represent a lack of respect."
The punishment is in addition to the one-match ban Messi received for a sending off in the third-place match against Chile following an incident with Gary Medel.
Messi was also docked $1,500 in June for criticizing match officials and the quality of the playing surfaces at the competition, which host Brazil won.
"We were ready for more, they did not let us get to the final," Messi said after the semifinal loss to Brazil. "Corruption and referees do not allow people to enjoy football and it ruined a little bit."
A three-month ban means the Albiceleste skipper will miss upcoming international friendlies against Chile and Mexico in September, and Germany in October. The 32-year-old's previous one-match ban is for Argentina's first 2022 World Cup qualifying match.