Spurs head coach Gregg Popovich made the announcement after Wednesday's shootaround.
Ginobili suffered the injury during the second quarter of Monday's loss to Minnesota after being called for a reach-in foul. He immediately grabbed his left hand in obvious pain.
"Manu is pretty important to us and we lost him," Popovich said. "We will just have to deal with it."
The oft-injured Argentine leads the Spurs in points (17.4), steals (1.2), field-goal percentage (59.2) and three-point shooting (51.9).
San Antonio recalled guard Gary Neal from a rehabilitation assignment with the Austin Toros of the NBA Development League and signed forward Ike Diogu on Tuesday to prepare for the loss of the two-time All-Star.
Neal spent last season with the Spurs, averaging 9.8 points, 2.5 rebounds and 1.2 assists through 80 games, while Diogu has appeared 223 career games with Golden State, Indiana, Portland, Sacramento and the Los Angeles Clippers, averaging 6.0 points and 3.1 rebounds.