Kvaratskhelia twisted his knee during PSG's 4-2 win on Tuesday and was helped off the pitch just before the hour mark, with the Georgian winger unable to put any weight on his leg.
The injury robs PSG of last season's best player in the competition as they sit top of this year's standings with a perfect record of three wins from three games as they seek a third straight title.
"Before every game I asked God to let me leave the pitch healthy. But it was in vain," Kvaratskhelia wrote on Instagram.
"Not much time has passed, yet one thought keeps circling in my head: why me? I never held anything back. I think I prepared for this game better than I ever had before. I thought I was ready for everything - but not for this.
"I know I have to come back stronger for my people, who are always by my side. I won't let you down - I promise I won't waste a single moment."
ACL injuries typically require a recovery period of at least six months.
The 25-year-old scored 10 goals and provided six assists as PSG defended their European crown last season, with his form also earning him a spot on the shortlist for the 2026 Ballon d'Or.
Dembele and Marquinhos also injured
Tuesday's win came at a heavy cost for PSG with reigning Ballon d'Or winner Ousmane Dembele limping off in the 19th minute with a thigh injury, while defender Marquinhos also came off with a similar problem.
Dembele's injury is another blow after the Frenchman began the season with four goals and six assists in all competitions.
PSG said both Dembele and Marquinhos will remain under treatment for the next few weeks.
Luis Enrique's side are second in Ligue 1 and next play on Sunday against table toppers Olympique Lyonnais, who are ahead on goal difference.
