What size ground wire do I need for a 600 amp service?
There are two different ground wires on a 600 amp service and they are sized from two different tables. Mixing them up is the most common error on this question.
The grounding electrode conductor
The grounding electrode conductor, or GEC, runs from the service disconnect to the electrodes. It is sized from Table 250.66 based on the largest ungrounded service entrance conductor, and where the service uses parallel sets you use the equivalent area of the parallel conductors per the note to that table.
A typical 600 amp copper service is two parallel sets of 350 kcmil, giving 700 kcmil equivalent per phase. Table 250.66 puts over 600 kcmil through 1100 kcmil copper at a 2 AWG copper GEC. A typical 600 amp aluminum service is two sets of 500 kcmil, giving 1000 kcmil equivalent, which lands in the over 900 through 1750 kcmil aluminum row and calls for a 2 AWG copper or 4 AWG aluminum GEC by that same row. Run your own conductor sizes through the table rather than trusting a rule of thumb, because 600 amp services get built with three different conductor combinations.
The exceptions that shrink it
Section 250.66(A) says that where the GEC or a bonding jumper connects solely to a rod, pipe, or plate electrode, that portion is not required to be larger than 6 AWG copper or 4 AWG aluminum. Section 250.66(B) caps the portion going solely to a concrete encased electrode at 4 AWG copper. Section 250.66(C) caps the portion going solely to a ground ring at the size of the ring conductor, which is 2 AWG minimum under 250.52(A)(4). So on a 600 amp service with a Ufer and two rods, the run to the rods can be 6 AWG copper even though the run to the concrete encased electrode is 4 AWG copper.
The equipment grounding conductor
If the question is really about a 600 amp feeder to a downstream panel, that is Table 250.122 and it is sized by the overcurrent device. At 600 amps the table calls for 1 AWG copper or 2/0 aluminum. Where the ungrounded conductors are upsized for voltage drop, 250.122(B) requires the equipment grounding conductor to be increased proportionally by circular mil area. Where the feeder runs in parallel raceways, 250.122(F) requires a full size equipment grounding conductor in each raceway, each sized to the full overcurrent device rating.
The main bonding jumper
Do not skip this one on the service. Section 250.28(D)(1) sizes the main bonding jumper by 250.102(C)(1), which uses the same 12.5 percent of the phase conductor area rule once the phase conductors exceed 1100 kcmil. On two sets of 350 kcmil copper the jumper follows Table 250.102(C)(1) and comes out at 2 AWG copper.
Field takeaway
Run the service conductor size through Table 250.66 for the electrode conductor, use the 6 AWG cap for a rod only run, and use 1 AWG copper for a 600 amp feeder equipment ground from Table 250.122.