The D600 is not on the list of supported cameras for 4.3 and is apparently not listed for 4.4RC, either. Ugh. Maybe in the final version of 4.4. I seem to recall that 4.3RC and 4.3 final were not identical in some surprising ways. However the D600 does tether to Camera Control Pro which, by default, hands the file off to ViewNX. I imagine you could send the files to a folder watched by Lightroom and use LR as the viewer. You can try using Sofortbild if you're on Mac. On the PC side, at least ControlMyNikon does not yet list support for the D600. There may be others that do.
That is Adobe's statement but only in re raw file support. This is a status change from 4.2 which had preliminary but not 'full' support of the D600 (see footnote at bottom of list). In other words, "4.2 should work but don't sue us if it doesn't". As mentioned, in the most current list of cameras supported for tethered capture, the D600 does not appear while the D700 (no surprise) and D800 do. See also this exchange at Adobe forums and pay particular attention to reply#2. If Victoria Bampton says it ain't so, it ain't so.
Yes, it does support raw files. Agreed. LR just downloaded and opened _DSC3048.NEF successfully. No, it doesn't support tethered capture. The implication that "fully supported" means that tethered capture is included within the meaning of "fully supported" in a thread dedicated to the D600 inability to tether to LR4 is unsupported. Please see my reply (#5) just prior. I regret if my statement referenced above was insufficiently precise as to which list it referred. In a fit of concision I omitted mention of 'tethered capture', hoping that the subject of the thread might carry understanding.