Both Tom Bombadil and the Nazgul are powerful magical beings within the context of the Lord of the Rings Universe. What would happen in a face to face fight between the two? Who would win?
As Gandalf put it in The Council of Elrond:
No, I should not put it so. 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.
A Nazgul, as a servant of the Ring, would therefore have no power over Bombadil, but Bombadil on the other hand would also be ineffective against a Nazgul. What would happen? Nothing.
Edit to add:
On double-checking the text, I've come across the following from Fog on the Barrow Downs:
Out east my knowledge fails. Tom is not master of Riders from the Black Land far beyond his country.
This seems conclusive; Tom couldn't defeat a Nazgul, but whether a Nazgul could defeat him depends on interpretation of Gandalf's statement above (with strong circumstantial evidence against).