"France will do all it can to find out the truth about what happened to Gilberto Rodrigues Leal and will not leave this crime unpunished," Hollande said in a statement.
Although Mujao announced that it had killed him because France "is our enemy", the government believes that Leal, a French citizen of Portuguese origin, died several weeks ago "due to the conditions of his captivity", the statement said.
The foreign affairs ministry said that it had warned his relatives that it feared he was dead in December.
Leal, who was retired, was abducted on 20 November on 2012 in western Mali.
On Sunday, after the liberation of four French journalists who were being held in Syria, Interior Minister Laurent Fabius said that he was "very worried" about Leal's situation.