He would have fallen, as stated in the Fellowship of the Ring:
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I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come.
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He would fall, indeed. I never said he is not subject to the fate of Middle-earth or natural needs such as sleeping or eating. But things don't influence him as they do others, he is not corrupted by the Ring, and so probably by nothing else.