Can we verify gap energy of a material by applying voltage across it?
Not that way.  

The bandgap can be measured with the voltage generated by controlled optical energy or thermal energy and is measured by the incremental slope of log saturation current for a fixed low reverse voltage for step increments of temperature.

You measure the diode reverse saturation current in uA in 10'C increments with a low reverse voltage from 0.2 to 2V.

You apply a small reverse voltage to achieve the constant saturation current vs temperature curve. One could heat up the diode in water from say 40'C to 70'C and plot the slope of log Is vs 5036/T ['K] to get bandgap  ∆E.   

log Is = Constant – 5036 ∆E/T 

or ∆E = slope of Log (Is)*T/5036  for T['K]