The Real Madrid player, 23, will miss Saturday's Champions League final against Atletico Madrid in Milan.
The French Football Federation said scans had confirmed Varane, who has won 29 caps for Les Bleus, would be out for up to three weeks.
Sevilla defender Adil Rami, 30, has been called up by manager Didier Deschamps as his replacement.
France play Group A rivals Romania in the tournament opener on 10 June.
France squad for Euro 2016:
Goalkeepers: Hugo Lloris, Steve Mandanda, Benoit Costil.
Defenders: Adil Rami, Laurent Koscielny, Eliaquim Mangala, Jeremy Mathieu, Patrice Evra, Bacary Sagna, Christophe Jallet, Lucas Digne.
Midfielders: Paul Pogba, Blaise Matuidi, Lassana Diarra, N'Golo Kante, Yohan Cabaye, Moussa Sissoko.
Forwards: Antoine Griezmann, Dimitri Payet, Anthony Martial, Kingsley Coman, Olivier Giroud, Andre-Pierre Gignac.
Reserves: Alphonse Areola, Hatem Ben Arfa, Kevin Gameiro, Alexandre Lacazette, Adrien Rabiot, Morgan Schneiderlin, Djibril Sidibe, Samuel Umtiti.
France defender Raphael Varane has been ruled out of the host nation's Euro 2016 campaign with a thigh injury.