The Serie A club said the 22-year-old, who has scored nine goals for them since joining from Ajax Amsterdam in the close season, had undergone tests after returning on Sunday.
"The exams have confirmed a... complete tear of the anterior cruciate ligament of the left knee," Napoli said in a statement.
Milik, whose scoring form has helped the team cope after the departure of Gonzalo Higuain, will be sorely missed.
Napoli are second in Serie A, four points behind Juventus.
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Milik was injured when fouled to earn a penalty converted by Robert Lewandowski in 36th minute for Poland's second goal.
Milik played on but was substituted at halftime.
(Writing by Brian Homewood; Additional reporting by Piotr Kwiatkowski in Warsaw; Editing by Ken Ferris)
