Does voltage play any role in choosing the appropriate wire gauge?
The simple answer is you use the wire that your local code and the device manufacturer specifies for the situation.



But I think you are asking about how wire sizes are chosen for a particular application/current/voltage combination:

When choosing wire, current dictates the size of the conductor and voltage dictates the insulation.

Current causes the wire to heat up due to resistance. Metal expands and contracts when heated and cooled. This expansion and contraction, if too large, can loosen connections. Loose connections increase resistance, cause more heating, and will eventually allow a gap large enough to cause an arc or a high enough temperature to ignite surrounding materials or melt the insulation and cause an arc. A larger conductor reduces the resistance, which reduces these temperature changes. Therefore, using a large enough conductor keeps the expansion and contraction under the level that electrical fittings can tolerate without failing.

Similarly, voltage causes arcing so the covering (insulation) has to be designed to prevent arcing at the rated voltage. Usually you'll see common electrical wiring rated for 600 volts.

Another reason to use larger gauge wiring is to prevent voltage drop on long runs. This is generally not a problem when running wire in a house, but detached structures are a common place to see larger wire used.