International law is, as usual, muddled.

• You write that sovereign Greece are the rightful owners. Well, the Mables did belong to Athens, and then Macedonia (Hellenic League), and then Rome, and then East Rome (Byzantium), and then the Ottomans, long before there ever was a Greek state. Plus a few more in between. Arguaby any one of those would have had a right to sell them.
  There is the concept of odious debt in international law, which says that debts of a despotic government does not bind a successor state. Attempts to apply this tend to be hotly argued, and the Ottomans were widely accepted as a government at the time, no more despotic than most others.
• The Elgin Marbles are part of the Greek, European and global cultural heritage. The 1970 UNESCO convention protects cultural heritage, but key provisions like Article 7(a) (prohibition of transfer) are not applied retroactively. How this interacts with Article 11 (transfer by occupying powers is illicit) would leave lawyers busy for decades or more.

International law is, as usual, muddled.

• Article 11 of the 1970 UNESCO convention makes the transfer of cultural artefacts by an occupying power illicit. But were the Ottomans an occupying power in the sense of the convention, or a multi-ethnic empire?
  The Marbles had been created by Athens at a time when there was no such thing as a sovereign Greek state. The Athenians would have rejected the notion of such a state unless it meant that they could dominate the other Greek cities. Sparta disagreed and they fought bitter wars. Athens subsequently became part of Macedonia (Hellenic League), and then Rome, and then East Rome (Byzantium), and then the Ottomans. Plus a few more in between. Without hindsight, it is difficult to call the Ottomans less legitimate than, say, Alexander the Great.
• Article 7(a) of the UNESCO convention, banning certain transfers, does not apply retroactively.
• There is the concept of odious debt in international law, which says that debts of a despotic government does not bind a successor state. Attempts to apply this tend to be hotly argued, and the Ottomans were widely accepted as a government at the time, no more despotic than most others.
• There is an ongoing debate, and possibly an ongoing change to customary international law, regarding looted art from colonies. Yet the Ottomans would have rejected any suggestion that they were a colony and not a sovereign empire.

Greece does argue it has a moral claim to the Marbles. So it becomes less a question of international law, and more one of persuasion.