Does the voltage affect the AWG wire size choice?
Voltage rating is a function of insulation quality rather than wire gauge. How close two conductors are of different potential (voltage) and what the dielectric or insulator is between them is what determines whether the cable can safely carry the voltage without a breakdown of the material or an arc.
Wire gauge is determined primarily by current. For example, a wire at only one volt, but carrying hundreds of amperes would need to be quite thick.
Conversely, a wire carrying 1000V but a miniscule amount of current can be a smaller gauge but would need insulation rated for 1000V, so the insulation would be thicker than the conductor.
At multiple kV, you have to start worrying about arcing through the air. At high voltages, you have to start spacing out the conductors. (There's a reason transmission lines are as far apart from each other as they are.) Humidity will also be a factor.
I'm hoping it was just an example in your question, but if you have to ask about wire gauge and insulation, you should definitely not be working with 60kV (or even household mains voltages).
If wires were perfect conductors (superconductors with no resistance whatsoever) then wire gauge might be a moot point. The reason that you size wires according to current is because electrical current flowing through even comparatively small resistance of a wire (compared to a load) still creates heat, which in turn can start fires or cause the wire to fail open. This is also why you should use larger gauge wires for longer distances, because the small resistance of a given length of wire is additive.
Edit:
Per comments, it's worth adding that even if a wire is sized correctly for current and isn't heating up, it can still cause a voltage drop due to its resistance — just like any resistor. (A voltage drop is a decrease of voltage due to dissipation of energy.) You may have an application where a larger gauge wire is desirable to counteract this. If your application is low voltage, such as with LED strips, outdoor low voltage lighting, etc., a voltage drop can be more problematic because it is a larger percentage of the intended voltage. In short, a 1-volt drop for a 5-volt application is significant while a 1-volt drop for a 240-volt load is rarely an issue. (The voltage drop will depend on the current as well as the resistance in the wire, E=IR.)
So for this reason, you may consider wire gauge as tangentially dependent on voltage as well, when you also factor in total wire length (resistance) and the specifics of the load.