Isildur didn't necessarily have to have any "Elven" powers.
In Anglo-Saxon society, the King/Leader was the ultimate authority. All warriors and citizens had to pledge fealty to him, and had to follow his every command. If they went back on their word, they would be considered traitors of the worst sort. Since Tolkien based much of Middle-earth on Anglo-Saxon society, I'm assuming that the same principles hold. Isildur had the right of the disobeyed leader to curse the traitors, and more often than not, when one has the right to do something, one has the power to do that "something." In this case, it was cursing.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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