Can I use #6 ground wire for 200 amp service?

Number 6 copper is acceptable on a 200 amp service only for the wire that lands on ground rods. NEC 250.66(A) caps the rod connection requirement at 6 AWG copper no matter how big the service is.

The rest of the grounding electrode system needs more. For a typical 200 amp service with 2/0 copper conductors, Table 250.66 requires 4 AWG copper to a metal water pipe or concrete-encased electrode.

So a code-built 200 amp service usually shows both: #4 to the water pipe, #6 to the rods.

Equipment grounds are a third, separate question: a 200 amp feeder's equipment grounding conductor is #6 copper per Table 250.122. Same size, different table, different job.

The 200 amp ground wire calculator sorts the three ground types so each run gets the right size.