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, the BBC reported.
Tensions between France and Turkey are high after President Emmanuel Macron pledged a tougher stance against radical Islam.
Erdogan has called on Turks to boycott French goods and said Macron needed "mental checks".
Macron pledged to defend secularism following the killing of a teacher who had shown his students a cartoon of the Prophet Muhammad.
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.
Presidential Communications Director Fahrettin Altun said: "Charlie Hebdo just published a series of so-called cartoons full of despicable images purportedly of our President. We condemn this most disgusting effort by this publication to spread its cultural racism and hatred."
Vice-President Fuat Oktay called on the international community to raise its voice against "this disgrace".
"You cannot fool anyone by hiding behind freedom of thought," he said on Twitter.
In response, the Turkish pro-government satirical magazine Misvak posted a number of cartoons criticising Macron and Charlie Hebdo on its Twitter page.
In 2015, 12 people were killed in an attack on the offices of the magazine in Paris. It was targeted by Islamic extremists for publishing cartoons of the Prophet Muhammad.
The same year, Russia heavily criticised the magazine for two cartoons depicting the Sinai air crash in which 224 people, mostly Russians, died.
In 2016, a cartoon depicting Italian earthquake victims as pasta dishes caused outrage.
Visit news.dtnext.in to explore our interactive epaper!
Download the DT Next app for more exciting features!
Click here for iOS
Click here for Android