Adnani, who was headmaster of the Lycée Bonaparte, accused the woman of not having degrees that she claimed to have, only to have her take legal action against him for his allegedly "anti-Muslim attitude".
At the beginning of September he was thrown into jail and, in what the French foreign affairs ministry calls a "compromise solution", quit Qatar on 8 September, leaving his family behind, Mediapart reports.
Although his name was on the headmaster's message at the start of the new term, Adnani has been offered another job in France, the ministry says, and the financial director has resigned.
Another French school in Qatar, the Lycée Voltaire, which was opened by then-president Nicolas Sarkozy in 2008, has already gone through two headmasters.
The first, Jean-Pierre Brosse, was recalled after Qatar's public prosecutor complained about textbooks that referred to the history of Christianity and France's wine production, along with a picture of a naked woman in a science manual, according to Le Nouvel Observateur.
A replacement, Franck Choinard, was employed at twice his predecessaor's salary but that did not stop the Qataris accusing the body that ran the school, the Mission Laïque, of siphoning off 500,000 euros to schools in other countries, forcing it to leave the emirate in 2012.
Choinard himself brought a case against the group in France but prosecutors have since dropped the case.
Soon after the new headmaster had to leave the country, accused of paedophilia.
He denies the charge and claims that it was brought because he refused to fire two of his staff.
While the Lycée Bonaparte caters mostly for expatriates' children, the Lycée Voltaire's pupils mainly come from rich Qatari families.
French President François Hollande visited the Lycée Voltaire on 22 June to mark an extension of its buildings.
Qatar has large financial interests in France, including hotels in Paris and Cannes and Paris Saint Germain football club.
France hopes to sell a number of Rafale fighter jets to the emirate's air force.