Cuche, 60, served as a commander of Nato-led forces in Kosovo in the 1990s. He resigns after two years as French army chief of staff.
Fifteen bystanders, including five children were wounded in the shooting, as were two soldiers.
A three-year-old boy was in serious but stable condition after taking bullets in the heart and in the arm.
The 28-year-old sergeant who fired the shots from his assault rifle was being held in custody and is expected to be charged on Tuesday with causing unintentional injury.
The man, who was officially suspended from duty on Tuesday, has been described as an experienced soldier with no history of psychological problems.
Prosecutors said he had been carrying a live cartridge in his pocket, wrongly left there after a recent military operation, and had used it to reload his rifle instead of the blank magazine.
General Cuche had ordered an army command investigation to establish the circumstances of the shooting, with a general due to report back within a week on the chain of responsibilities.
But Herve Morin, France's defence minister, contacted Cuche to ask for immediate sanctions, saying the "tragic accident" had revealed "serious shortfalls" in army safety procedures.
"The minister asked the army chief of staff, without waiting for the outcome of the judicial and army command investigations, to put forward... immediate sanctions for those involved in the dysfunctions observed," a defence ministry statement said on Tuesday.
After visiting the wounded children in hospital on Monday, Sarkozy called the shooting the result of "unacceptable negligence", for which he promised a "rapid and severe response".
Sarkozy's office said he was "closely following the various investigations" and that "he intends for the armed forces to draw full consequences in terms of organisation and operations."
A witness, who asked not to be named, said he saw seven or eight soldiers with guns taking part in the simulation, one of whom was posing as a terrorist in the middle of the spectators.
"Suddenly, people were falling, we thought it was part of the exercise, and then we saw blood."