The Barcelona and Argentina captain said his side had suffered “corruption” at the tournament in Brazil after being sent off in a third-place playoff.
South American football confederation CONMEBOL also handed the five-time Ballon d’Or winner a 50,000 dollar (£41,000) fine.
The ban means La Albiceleste will be without Messi for friendlies against Chile, Mexico and Germany.
The Barcelona star was dismissed in the Copa America following a confrontation with former Cardiff midfielder Gary Medel, who was also shown a red card.
Messi appeared to have been hard done by, with Chile defender Medel appearing to push his head towards the Argentina playmaker as they squared up, chest to chest.
The Argentina captain suggested he had been singled out following his comments over the “crazy” officiating in the wake of their 2-0 semi-final defeat by hosts Brazil in Belo Horizonte.
We don’t need to be part of the corruption that we’ve suffered at this tournament,Messi said, as quoted by Spanish media outlet AS.
“Medel is always right at the limit. With a yellow that would have been the end of it for both of us, but well, maybe what I said recently had an impact.
“What’s important is that the team finished well in the tournament. Maybe this was ordered and I ended up suffering because of what I said.”
In the aftermath, CONMEBOL said a “fundamental pillar of fair play is to accept the results with loyalty and respect”.
Although it did not mention Messi by name, the statement issued at the time said: “The same goes for refereeing decisions, which are human and will always be improvable.”