Pato was carried off injured in the 70th minute of the goalless draw against Udinese after falling awkwardly in a challenge with defender Christian Zapata.
He will be in plaster for at least a month, Milan club doctor Maurizio Gevi said.
"He will then have to have rehabilitation," Gevi told reporters. "He won't be going to the Copa America. His season is over."
After an injury-plagued 2010, 21-year-old Pato had returned to top form for his club this year, scoring a number of crucial goals to help Milan win the Serie A title for the first time in seven years.
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