In this question, we've learned about the difficulty for Elrond and/or Círdan to do anything amounting to taking the ring from Isildur, once the latter decided it was his loot. It seems they tried to convince Isildur to destroy the ring, acknowledging him to be the de jure owner of the object.
But how was it established that it was legal or natural for Isildur to become the new owner of the ring? Did not Elendil and Gil-Galad jointly incapacitate Sauron in duel? Did not Isildur simply step forth to an unconscious Sauron and cut the ring from his finger. Perhaps Isildur was just lucky to stand closer to the spot of the duel than Elrond & Círdan, thus allowing him to make that cut.
In his written account, Isildur seems to 'blow up' the importance of his cutting the ring from Sauron's hand (" Was it not I that dealt the Enemy his death-blow?") and he adds that he lost his father and his brothers in the war and would keep the ring as a 'weregild'.
Of course, the elvish party in the war also suffered many casualties (although Gil-Galad, Oropher,... were not directly related to either Elrond or Círdan, the latter may have loved Gil-Galad or other deceased members of the last alliance in the same way as Isildur loved his kin). If Elrond and Círdan had such a high stake in getting the ring destroyed: why did they not challenge Isildur's legal right to own the ring?
1) I don't think that Isildur's claim and decision to keep the ring was a heat-of-the-moment event or decision. It is said in various places that the armies of the last alliance stayed in Mordor for some longer time to attempt to dismantle Barad-Dûr and Sauron's realm at large. I can imagine the Númenorean and elvish parties therefore remained in contact for some time after Sauron's fall and Isildur had plenty opportunity to reconsider/repent and there must have been sufficient time to raise questions of legality (and the elves may have urged him to?)
2) The "Disaster of the Gladden Fields" chapter of the Unfinished Tales suggests that, if Isildur would not have been ambushed in the Gladden fields, he would have gone to Rivendell to surrender the ring to "the Keepers of the three":
"It needs one greater than I now know myself to be. My pride has fallen. It should go to the Keepers of the Three." (Isildur to Elendur)
(He attributes his change of mind not to any legal matters though. He rather states that "...I cannot use it. I dread the pain of touching it"). That chapter is however conceived by Tolkien as Gondorian in-universe-lore originating from eye-witness accounts of the Gladden fields disaster. One can imagine that the story was modified over the course of the Third and Fourth age to give an account where Isildur's conduct and thought is whitewashed.