I think what it comes down to, as with many versus battles involving Kryptonians, is speed.
IMO the Destroyer is offensively powerful enough, in particular his disintegration beam, to injure or even kill Power Girl, but he has to be able to hit her to do so. Given her vastly superior speed and reflexes this should be almost impossible.
Now is Power Girl... er powerful enough to damage the Destroyer? The Silver Age character is a yes and even modern incarnations should be able to do some damage. Again, with her speed she can always throw the damn thing into space or the sun.
I'd expect Power Girl to win 9 times out of 10 battles losing only if she is unlucky and gets tagged unexpectedly.
Note: I accidentally voted for the Destroyer, but I meant Power Girl.