The incident occurred with 7:17 remaining in Thursday's game at Cleveland. The pair became tangled under the basket and went face-to-face before White knocked heads with Ilgauskas.
The 6-9, 290-pound White did not play in Friday night's home game against the Atlanta Hawks.
Ilgauskas was fined $5,000 for his part but was not suspended.
However, Cavaliers rookie center DeSagana Diop was banned one game without pay and was fined $5,000 for leaving the bench to aid Ilgauskas. The 7-footer was knocked to the floor by Cavaliers assistant Keith Smart before reaching the skirmish.
Celebrity Families of 2014 [PHOTOS]