Does the diameter of the coil in a motor determine the amount of volts needed to run it?
No. It's a mechanical design consideration; indirectly, it does affect field strength, but that only indirectly links with voltage...
So, Motor design is a complex topic full of tradeoffs.
There's literally dozens of different kinds of eletrical motor designs, and every one of them has a different relation between coil diameter, wire resistance/diameter, coil currents, forces and moments, speed and voltages.
I'm afraid you'll have to dig through a bit of literature on your specific motor type if you need info on how to build a motor to operate at a certain voltage giving you certain operational characteristics (current, torque, speed, reluctance, temperature…).