The 27-year-old was withdrawn after colliding with Guingamp striker Mustapha Yatabare during the first half of an eventual 1-0 victory at the Stade Velodrome on Saturday.
Mandanda had been named alongside keepers Hugo Lloris of Tottenham and Bastia's Mickael Landreau in France's 23-man squad for the Brazil finals, and his place has gone to Stephane Ruffier of Saint-Etienne.
Club doctor Chriostphe Baudot said: "Steve Mandanda is suffering from a crack in his first cervical vertebra and from a cervical sprain.
"It will require wearing a neck brace for three weeks and the same amount of time in physiotherapy."
Marseille coach Vincent Labrune added: "I feel a great disappointment for the player, who is going to miss an important competition."