Elgin Marbles: no litigation after all, but diplomacy to continue

"Shall we go back to Athens, then?
It's positively 'friezing' here in London ..."

"Elgin Marbles legal action ruled out by Greece" is the news from the BBC website today. The article, by an unnamed author, can be read here. It states, in relevant part:

Greece has ruled out taking legal action against the UK to reclaim the Elgin Marbles from the British Museum. In an unexpected move, Greece's culture minister said the country would pursue a "diplomatic and political" approach to retrieving the sculptures instead.

In doing so, the country has rejected the advice of barrister Amal Clooney, who had urged Greece to take Britain to the International Court of Justice.

Lord Elgin acquired the Marbles from the Ottoman Empire 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 Mrs Clooney. Mrs Clooney reportedly submitted a 150-page report to the Greek government this week urging it to formally request the repatriation of the marbles and take Britain to the International Court of Justice if it refused. 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. ...

An earlier initiative to have the dispute mediated by UNESCO was rejected by the British (on which see Marian's post here). This blogger was sad about this, since mediation never hurts: a mediator cannot impose an outcome on an unwilling party and the process itself often enables both sides to be more sensitive to the interests and the anxieties of the other.

This blogger is however curious to know the precise legal basis upon which a legal claim for repatriation of the Elgin Marbles would be made, and whether the same argument would effectively provide a basis for repatriating quantities of other museum holdings. Can any reader advise?