Saliba was hampered by the injury throughout the tournament in the United States, limiting his training, before he was substituted after 30 minutes of France's semi-final defeat by Spain and missed the third-place playoff loss to England.
"Subsequent specialist reviews on William's return to London this week have confirmed that he has suffered a back injury which will require a period of rehabilitation," Arsenal said in a statement.
"Extensive assessments have concluded that surgery is not recommended, but William should now commence a managed recovery programme.
"Everyone is fully focused on supporting William to ensure he is back to full fitness as soon as possible."
Arsenal begin the defence of their Premier League title at home against promoted Coventry City on August 21.