Greece's culture minister said the country would pursue a "diplomatic and political" approach to retrieving the sculptures from the British Museum, instead.
Greece rejected the advice of barrister Amal Clooney, wife of Hollywood star George Clooney, who had urged Greece to take Britain to the International Criminal Court.
The Marbles were taken from Greece's Parthenon by Lord Elgin 200 years ago.
Greece insists the Parthenon Sculptures, as they are properly known, were taken illegally and has pursued a high-profile campaign in recent years for their return, latterly with the help of human rights lawyer Clooney.
Also Read
But Greece's culture minister Nikos Xydakis told the country's Mega TV: "One cannot go to court over whatever issue. Besides, in international courts the outcome is uncertain."
He said he believed attitudes to the future of the Marbles were slowly changing and would favour Greece in a diplomatic approach.
For 30 years, Greece has been locked in a bitter dispute over its demand for the marbles to be returned.