The cartoon depicts Turkey’s president lifting the dress of a veiled woman.
State media say Turkish prosecutors have launched an official investigation into the satirical magazine.
Tensions between France and Turkey are high after President Emmanuel Macron pledged a tougher stance against radical Islam.
Mr Erdogan, who has also launched legal action against an anti-Islam MP in the Netherlands over a separate cartoon, has called on Turks to boycott French goods and said Mr Macron needed “mental checks”.
The dispute has reverberated across the world, spurring boycotts and protests against France in several Muslim-majority countries including Bangladesh, Kuwait, Jordan and Libya.
The controversy also follows Mr Macron’s pledge to defend secularism after the gruesome murder of a French teacher who showed cartoons of the Prophet Muhammad in class.
The president said the teacher, Samuel Paty, “was killed because Islamists want our future”, but France would “not give up our cartoons”.
Depictions of the Prophet Muhammad are widely regarded as taboo in Islam, and are offensive to Muslims.
But state secularism is central to France’s national identity. Curbing freedom of expression to protect the feelings of one particular community undermines unity, the state says.
Source : BBC News