D600 with 24-120mm AF-S NIKKOR f/3.5-5.6 VR?

No, it's quite good on both my D800 and D800E.

Therefore, I think it would also work well with the D600.

Have a look at Michael Reischmann's review of the D800 on Luminous Landscape where he describes his lens choices. The 24-120 makes the grade very favorably.