The 32-year-old was injured during a collision with Wolfsburg goalkeeper Koen Casteels on Saturday as Stuttgart ran out 1-0 winners in the Bundesliga clash.
Gentner was forced to leave the pitch at the time which left the side down to ten men as they had used all their substitutions, but they held on to seal the three points.
Afterwards, head coach Hannes Wolf, who does not yet know how long Genter will be sidelined for, said: "The game was overshadowed by the situation with Christian.
"We were all in shock and are thoughts are now with our captain. We hope he'll get well soon."