hmm...according to this you do need a permit.
You will need a recreation permits for most of the Apache Nations such as the following San Carlos Tribal Government rates: ($10 per day) to camp, picnic, hike, or venture onto back roads unless you have a fishing, hunting, or special-use permit.
The closest one to the route would be the Apache Gold Casino
I've visited the dam (took Coolidge Dam Rd. all the way) and I didn't have a permit. Maybe I just got lucky.