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Brazil team doctor Lasmar, who carried out the surgery, said the 31-year-old is not expected to be fit until August and they do not intend to take short cuts in the recovery process for him to play at the June 20-July 14 tournament in the US.
The top two in each of the four groups proceed to the quarterfinals.
The June 20-July 14 tournament features 32 matches and 16 participating teams - 10 from CONMEBOL and six from CONCACAF.
The eight-time Ballon d'Or winner has bittersweet memories of the last time the Copa America was held on American soil in the 2016 centenary edition, where Argentina reached the final but lost on penalties to Chile.
Di Maria has made 136 appearances for world champions Argentina since his debut in 2008, playing in four World Cups and scoring in the 2022 final win over France.
The announcement comes after both European governing body UEFA and their South American counterparts CONMEBOL came out strongly against FIFA's plans for a biennial World Cup.
Hernan has followed Messi from the very beginning of his career. He has recorded, by hand, with a pencil in his notebook, every single goal Messi has scored since starting his career.
The Argentine talisman is set to take a wage cut to continue with the La Liga giants
The duo was together at Barcelona for four years before Neymar moved to Paris Saint-Germain.
Scaloni did not say what the injury was but he showered Messi with praise on a night Argentines took to the streets in huge numbers to celebrate their first international title in 28 years and Messi’s first victory with the national side.