Caleb Moore was stopped on suspicion of speeding shortly after 4 p.m. Sunday in Crenshaw County.
Trooper spokesman Kevin Cook said he was charged with driving under the influence, possession of a controlled substance, possession of drug paraphernalia and speeding. He was being held Monday on $8,500 bond.
Roy Moore said he couldn't discuss details because his son had applied to be treated as a youthful offender.
Roy Moore is president of the Foundation of Moral Law and has formed an exploratory committee to run for president. He was removed from office as chief justice for refusing to move a Ten Commandments monument from the court lobby.