There is evidence that Tom Bombadil is unaffected by the One Ring's power and he has magical ways about him that can't be explained.
It seems that Tom did not participate actively in the War of the Ring, even though he was clearly more powerful than others.
Is it possible that Tom Bombadil could have used his magic to defeat Sauron?
No, Tom could not have defeated Sauron. As Gandalf mentioned, Tom could not use the Ring's power - rather the Ring did not affect him.
'It seems that he has a power even over the Ring.' 'No, I should not put it so,' said Gandalf. `Say rather that the Ring has no power over him. He is his own master. But he cannot alter the Ring itself, nor break its power over others.
Galdor and Glorfindel then describe Bombadil as not being able to withstand Sauron in the end, once Gandalf suggests he won't lead the use of the Ring:
'But in any case,' said Glorfindel, `to send the Ring to him would only postpone the day of evil. He is far away. We could not now take it back to him, unguessed, unmarked by any spy. And even if we could, soon or late the Lord of the Rings would learn of its hiding place and would bend all his power towards it. Could that power be defied by Bombadil alone? I think not. I think that in the end, if all else is conquered, Bombadil will fall, Last as he was First; and then Night will come.'
'I know little of Iarwain save the name,' said Galdor; 'but Glorfindel, I think, is right. Power to defy our Enemy is not in him, unless such power is in the earth itself. And yet we see that Sauron can torture and destroy the very hills. What power still remains lies with us, here in Imladris, or with Cirdan at the Havens, or in Lórien. But have they the strength, have we here the strength to withstand the Enemy, the coming of Sauron at the last, when all else is overthrown?'
Both quotes from the Fellowship of the Ring.
If this is a question of who is more powerful it is an unanswerable one. The origins and powers of Tom Bombadil are far too ambiguous to be matched up against Sauron with a declared victor. If this is a question of whether or not Tom ever would the answer is simple: no. Tom is uninterested in the affairs of the ring or the world outside the Old Forest. He would not seek to stand against Sauron whether or not he lurked around the Old Forest's borders or not.
No, he could not have, because Tom Bombadil is a literary device more than an actual character in the books. He exists in order to relay information which Tolkien must have decided would not reasonably be available otherwise, and in a form, or tome, or style, which is reminiscent of literary work by Tolkien outside of the Lord of the Rings, so it merits strong separation from the properly-in-world characters. Thus it doesn't make sense to really get him involved in world events (by challenging Sauron or any such action).
Sauron is all-powerful. Unless he is destroyed he is unbeatable and unstoppable. The destruction of the ring (and Sauron's native power which is VERY considerable) is the only way to defeat him. Tom Bombadill is not so much powerful as outside the power game that is being played out in the war of the ring. He has his own agenda and purpose (under Eru). However he cannot stand once the very fabric of Middle Earth has fallen to Sauron.
