Groves suffered the damage above his right eye in the third round of his stoppage victory over Mexican Francisco Sierra in San Jose, California in the early hours of Sunday morning.
The 24-year-old dispatched Sierra in the sixth round, but the cut will prevent him from defending his British and Commonwealth titles against Anderson at Wembley Arena.
It is the latest injury setback for Groves, who until Sunday had been inactive since November because of back and nose injuries.
Following Groves' withdrawal the September 14 event, promoted by Frank Warren, will now take place at York Hall
Billy Joe Saunders has been promoted to the top of the bill with a mandatory defence of his Commonwealth middleweight tile Championship against unbeaten Australian Jarrod Fletcher.