Oh boy. Here we go:

Elgin purchased the Parthenon sculptures from the Ottoman Turks, who at the time of the purchase were the rulers of Greece. At the time, it was a legal sale. You can imagine what subsequent, post-Ottoman Greek governments have to say about the legitimacy of that transaction.

There is no doubt that in the early years of their being in London the British treated the marbles better than they would have been in Greece. (At the time Elgin exported the marbles, other traders had set up lime kilns on the Acropolis to render marble into lime/white wash. The permits for the kilnsmen are documented. Without Elgin, in other words, the Parthenon marbles would almost certainly be dust.)

Among the claims made by Britain to justify the retention of the marbles is the assertion that the sculptures have been in London for so long, they are now to be consider part of the British heritage. As well as Greek heritage, which as a Western democracy, Britain also claims as its heritage.

Greece claims ownership based on initial creation and on the fact that they have the building on which the marbles should historically be hung. However, no one is planning to restore the marbles to their original on site location. So the question becomes whose museum they should occupy. The Greeks counter British arguments about appropriate curation/care by claiming—with full legitimacy—that whatever might have happened in the past, the marbles would be well treated in Greece today. Indeed, the recently opened new Acropolis museum has a top floor gallery specifically designed to hold the sculptures when(ever) they actually find their way "home."

Neither side is willing to budge on the issue at the moment, and currently the Greeks have other things to worry about.

As for the Native American artifacts, repatriation of those to the tribes is usually covered in the U.S. under the Native American Graves Protection and Repatriation Act, which focuses on ownership based on tribal affiliation and descent. When the Smithsonian gives stuff back to the Indians, there is a recognized lineage claim. In some instances, this claim has been stretched (for example, Kennewick Man).

Claims to ownership of Greek and Roman Mediterranean antiquities are usually today based on alignment with the 1970 UNESCO convention on the Means of Prohibiting and Preventing the Illicit Import, Export and Transfer of Ownership of Cultural Property 1970. If the piece can be documented as being owned before 1970, most scholars and countries will allow the ownership to stand. If there is no documentation of legal export before that date, the ownership is often in dispute and many of the Western scholarly associations (for example, the Archaeological Institute of America) will not allow their conferences or their publications to serve as the initial point of publication/announcement of the item.