When he made the Ring, Sauron poured a lot of himself into the Ring, which is what gave it its "malice" and "cruelty." He was only defeated when the Ring was taken from him, effectively separating "parts" of himself. When the Ring was destroyed, Sauron lost a great deal of "himself," which had been put into the One Ring. He couldn't survive in any form or manner with such a big "chunk" of himself missing like that.
To put it in human terms, we wouldn't be able to live without arms and legs, right? Sauron couldn't either.
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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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