The star says such images put his two children in danger, particularly due to his wife's line of work.
Amal Clooney is a human rights lawyer who, the actor notes in his letter, puts terrorist groups on trial.
His comments were published by several trade magazines, including Variety, The Hollywood Reporter and Deadline.
The actor, who has starred in Gravity and Ocean's Eleven, said he and his wife are not on social media and are protective of their children's privacy.
