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.

Henrietta Yakubu, a spokesman for the aviation authority, said repair works were completed 24 hours ahead of schedule.
An Ethiopian Airlines flight was the first to land on Tuesday morning.
Flights had been diverted to Kaduna, 160 km (100 miles) from the capital, and some international carriers refused to operate the new route.
Those airlines are now taking bookings for flights later this week.
The runway at Nnamdi Azikiwe airport was supposed to be upgraded in 2002 - it was built in 1982 and was only meant to have a 20-year lifespan.
There have been celebrations that the repair project was completed ahead of time but the fact the capital was without a functioning airport for more than a month highlights the state of Nigeria's crumbling infrastructure, the BBC's Martin Patience says.

Launched in February, the Hitomi X-ray satellite began tumbling out of control in March when contact was finally lost.
Just before its demise, scientists managed to extract data measuring X-ray activity in the Perseus galaxy cluster.
Published in Nature, data revealed the movement of gas between galaxy clusters was not as turbulent as expected.
Hitomi, which translates as the pupil of the eye in Japanese, was meant to spend years studying the formation of galaxy clusters and the warping of space and time around black holes.
It cost more than a quarter of a billion dollars - the research was an international collaboration involving the American space agency Nasa, and teams in Japan and many other countries, including one at Cambridge University in the UK.
Scientists studying the data found that the hot gases between galaxies within the Perseus cluster was travelling at a slower speed - at 540,000 km/h (340,000 mph) - and in a less turbulent way than many expected.
Measuring the turbulence and movement of such gas is important because the size of galaxy clusters is a useful tool for measuring "the parameters of cosmology and the growth of structure in the universe", researchers say.
The gases in galaxy clusters are thought to be particularly hot and dense with  a lot of dark matter and a supermassive black hole at the centre of it all. The interaction between all the various gases was thought to be very chaotic so Hitomi's observations appear to have defied predictions.
"The gas is relatively stable and isn't getting pushed around as much as we thought," said Brian McNamara, a professor of physics and astronomy at the University of Waterloo in Ontario, who was part of the research team.
"Hitomi's Perseus observation tells us that we can probably weigh distant galaxy clusters to greater accuracy than we can weigh our own Milky Way galaxy."
Hitomi was lost thanks to a sensor incorrectly detecting a roll in the spacecraft. In trying to correct it, on-board systems sent the craft into a spin until finally the solar panels that powered it are thought to have broken off.
The next time a similar satellite will be launched is in 2028 by the European Space Agency.
France defender Raphael Varane has been ruled out of the host nation's Euro 2016 campaign with a thigh injury.
[NEXT_CONCEPT]
Nigeria's airport in the capital, Abuja, has reopened six weeks after it was closed for urgent repairs.
[NEXT_CONCEPT]
A doomed Japanese satellite managed to capture a view of a galaxy cluster 250 million light years away just before it died, scientists have revealed.