Turkey's foreign ministry had already summoned Greece's ambassador over the headline in Dimokratia newspaper, also available on its website.
The headline "Siktir Git Mr. Erdogan", which means "fuck off" in Turkish, appears next to a photo of the president in the Greek newspaper, which also added the English translation.
"On behalf of the Turkish government, I condemn in the strongest terms the publication of insults directed at our President ... on the front page of an extreme right newspaper," Fahrettin Altun, communications director at the Turkish presidency, wrote in a letter to Greek government spokesman Stelios Petsas made public on Saturday.
He urged Greece to bring to account those responsible for the "shameless" action.
"Insulting a foreign leader is nothing but a sign of helplessness and lack of reason, and does not fall within the scope of press freedom or freedom of expression," he said.
The Greek foreign ministry had said on Friday freedoms of expression and the press were fully protected in the EU member country but added: "The use of offensive language is contrary to our country's political culture and can only be condemned."
The two NATO countries are already locked in a heated dispute over energy exploration in contested waters in the eastern Mediterranean.
Turkey's deployment last month of research vessel Oruc Reis prompted the neighbours to stage rival air and naval drills in strategic waters between Cyprus and the Greek island of Crete.
Last weekend, Ankara pulled the vessel back to the shore for maintenance and replenishment after its one-month mission.
But Turkey extended the operations of another research vessel, Barbaros Hayrettin Pasa, off Cyprus until October 18, according to a message on Friday on maritime alert system NAVTEX.
Turkey's drill ship Yavuz will also remain off Cyrus until October 12.
Erdogan said on Friday he wanted to give diplomacy a chance with Oruc Reis pulled back on the port and did not rule out a meeting Greek Prime Minister Kyriakos Mitsotakis via videoconference or in a third country.
French President Emmanuel Macron, whose country backs Greece in the dispute, welcomed the Turkish leader's willingness for dialogue, saying southern European leaders had delivered a similar message to Turkey during a meeting in the Corsican city of Ajaccio last week.
"In Ajaccio, we sent a clear message to Turkey: Let's reopen a responsible dialogue, in good faith, without naivety. This call is now also that of the European Parliament. It seems to have been heard. Let's move on," he tweeted in French and Turkish.
EU leaders are due to discuss possible sanctions against Turkey at their meeting on September 24-25.Follow us on Facebook, Twitter and Instagram to stay connected