Galtier, 57, won Ligue 1 with PSG last season but was replaced by former Barcelona boss Luis Enrique in July.
The Frenchman is due to stand trial in December, following an investigation into alleged racism while in his previous post as manager of Nice.
He was questioned by police in June and has denied the accusations.
Galtier replaces former Chelsea and Argentina striker Crespo at the helm, after the 48-year-old parted ways with Al-Duhail on Wednesday.
Frenchman Galtier has been named Ligue 1 manager of the year three times and has been a championship winner with Lille and PSG.
The Al-Duhail post is his first managerial job outside of France and he inherits a squad that features former Liverpool and Aston Villa midfielder Philippe Coutinho.
Galtier is the latest high-profile name to move to the Middle East from Europe, with many heading to the Saudi Pro League.
He will lead his new team against Saudi Arabian side Al-Nassr, who include the likes of Cristiano Ronaldo and Sadio Mane, in the Asian Champions League group stage later this month.
