In a statement, Italian champions Juve said that the 21-year-old had undergone "perfectly successful" surgery on the anterior cruciate ligament and meniscus in his left knee.
"The time required for an optimal recovery is 6-7 months," the club added.
Demiral limped off a quarter of an hour after opening the scoring in the 2-1 win at Roma on Sunday that put his side top of Serie A, damaging his knee after jumping for a header.
His probable absence from the Euros will be a blow for Turkey, who came to rely on Demiral during their successful qualifying campaign for this year's tournament.
Demiral has played 11 times for his country and appeared in every one of their qualifiers.
Turkey take on Italy in the opening match of Euro 2020 at the Stadio Olimpico in Rome on June 12.
Demiral was one of the Turkey players who celebrated the winning goal in his country's 1-0 qualifying win over Albania in October with a military salute in support of the Turkish offensive in Syria.
He repeated the gesture during their 1-1 draw with France a few days later.