"Sarkozy was unable to free his compatriot through this failed operation but he definitely opened one of the gates of hell on himself, his people and his nation," he said.
"As a quick response to the despicable French act, we confirm that we have killed hostage Germaneau in revenge for our six brothers who were killed in the treacherous operation," the voice in the tape, which resembled that of other recordings attributed to Abdul-Wadud, said.
Germaneau was kidnapped in northern Niger on April 20.
French commandos, acting with Mauritian troops, had tried to free him but did not find him when they raided a desert al-Qaeda camp in Mali, Sarkozy said.
He said France had received no sign since May that Germaneau was alive, and had intervened after AQIM threatened on July 11 to kill him within two weeks unless Paris arranged a prisoner exchange.
French special forces and Mauritian troops launched the cross-border raid against an AQIM base in Mali on Thursday and continued the assault for several days.
Mauritanian officials said six fighters were initially killed and a seventh later died of wounds.
The military operation appeared to anger Mali, which was not involved, and Spain, which also has hostages held by another al-Qaeda faction in the region.
Paris has indicated that both those countries were informed in advance.
A month after his kidnap, Germaneau's abductors issued a photograph of him as well as a taped message in which he appealed to Sarkozy to work for his release.
He said he suffered from a serious heart illness and had no more medication and that he was struggling with the heat.
Germaneau's Algerian driver, who was also abducted, was later released. He said the Frenchman was being held in a desert zone in Mali.
Current and former security officials in Algeria, which has the most experience with AQIM because it grew from a group that originated there, had warned that the raid could put the French hostage in danger and criticised French involvement.
One Algerian security official said the operation would help the fighters recruit more followers by allowing them to cast their campaign as a battle against Western "infidels" and not just fellow Muslims.
"The failure will be used by the extremists to spread their anti-Western propaganda," the security official, who did not want to be identified because he was not authorised to speak to the media, told the Reuters news agency.
A former Algerian security official said: "France failed to release its hostage. It failed to eliminate Abu Zeid....
"It angered the terrorist group, which will now either demand a ransom or kill the hostage if it has not done it already."
AQIM killed British captive Edwin Dyer last year after Britain refused to give in to its demands.
Last April, French troops attacked pirates who seized a yacht off the coast of Somalia. One hostage died and four were freed.