"I have made a decision to temporarily strengthen the French military presence in the Eastern Mediterranean in the coming days, in cooperation with European partners including Greece", Macron, who also declared Turkey's seismic work as illegal, said in a series of tweets on Wednesday night. Two French Rafale fighter jets are making a stop Thursday at Souda, on Crete, after being previously deployed in Cyprus for an exercise.
The French frigate La Fayette has been on exercises with the Greek navy and is staying in the area.
Turkey said the deal with Egypt - which Greek diplomats said effectively nullified an accord between Turkey and the internationally recognised government of Libya - showed it could not trust Athens, and vowed to continue surveying waters that are also claimed by Greece and Cyprus.
Mr Macron told Greek Prime Minister Kyriakos Mitsotakis that the French military would monitor the situation.
Mr Macron tweeted: "I have chose to strengthen the French military presence temporarily in the Mediterranean, in co-operation with Greece and other European partners".
Turkey and Greece, NATO allies, are vehemently at odds over claims for hydrocarbon resources in the region, and tensions have risen since Ankara launched exploration operations in an area of the Mediterranean on Monday. France had earlier criticised "aggressive" action by Turkish ships against a French vessel participating in a North Atlantic Treaty Organisation mission in the Mediterranean.
Erdogan said: "We don't have designs on anyone's rights, but we won't let any country take away our rights".
Turkey has said it plans to issue gas exploration and drilling licences in the region, somewhere between the Greek island of Crete and Cyprus, this month.
He also accused France of "provoking" Greece and Cyprus into taking "wrongful steps".
Turkey's president added that he would speak to German Chancellor Angela Merkel, who is the mediator between the two countries, and President of the European Council Charles Michel later on Thursday to discuss the issue.
The EU said the deal was a violation of worldwide law that threatened regional stability.
"The prime minister said Greece "(does) not fear even the toughest dialogue" as it had faith in its positions, but that "dialogue becomes irrelevant in a climate of tension and provocation".
US Secretary of State Mike Pompeo arrived in Prague on Tuesday (11 August), public TV footage showed, to open a five-day visit to central Europe with a hefty agenda including China's role in 5G network construction.
Athens has demanded the immediate withdrawal of the Oruc Reis from what it regards as its continental shelf, and has asked for an emergency meeting of European Union foreign ministers on the issue.