Does a higher gauge electrical wire impact voltage level?

Hey Emerging, not if there's no current on the circuit. :)  What you are talking about is commonly called "voltage drop."  Meaning, how much voltage drops on one end of a conductor compared to the other end, at the source.

Voltage drop is proportional to current in this formula:

V = I (current) * R (resistance)

That's the formula, but I is in Amps. So, no current, no drop. If the voltage at an outlet with no other current on the circuit is going up and down then it's caused at the source.  If the voltage goes up and down based on devices you turn on, then it's caused by wiring between you and the transformer.  In this case, higher gauge wiring would have lower drop because R would be lower.

Higher gauge means 4,6,8,10,12,14 ... wire diameter decreases as gauge number goes up. Resistance (R) measured in ohms increases as wire diameter decreases and gauge number gets higher.

The higher the gauge number the more resistance it has.

@erik_squires : your last sentence is incorrect!