Can we find potential difference of a P-N junction by connecting voltmeter?
The correct way to measure a diode is with a I-V curve tracer. The barrier potential occurs at the "knee" voltage in the forward bias. This is the point at which current increases greatly compared to the applied voltage.



I found a video of how to do this cheaply. 2 If you are student your electronics lab may have something like a Tektronix 370A somewhere you can use.

If you are wondering "why" this is you are asking a physics question and the answer is that the barrier potential is not really a "voltage". A diode is still a passive device like a resistor. You have to apply a voltage/current to get a meaningful output. 3