voltmeter and other voltage measuring equipments generally apply a specific voltage and measure voltage using the current flowing the resistor or coil connecting the voltage applied
or
the voltage to be measured is applied to a coil and using the current flowing through it the voltage is measured....
for the first case the applied potential has to be low enough not to disturb the built in potential for the second case the built in potential is the formed to stop the diffusion of carriers and so it doesnt supply current...
The barrier potential is formed because of immobile ions piling up on either side of the PN "Junction". This potential is more statistical and any absolute measurement is not possible...many factors affect the junction potential at a given instant...like temperature, doping, biases applied...etc....
I assume I have digital voltmeter which has infinite input impedance also
even if the built-in potential can't be known accurately its average is NOT zero
To make the picture more clear: When you draw the famous graph of
the electric potential vs. distance you will find that one terminal have higher potential than the other this means that diode @zero current have a finite potential
which contradicts I-V characteristics of diode....
many of the graphs taught are just approximations to help us in easier analysis... so generally some variations are to be accounted by deeper knowledge...
Between the p and n part of the diode there is a potential difference. you could also ask a similar question: why is there no current flowing when making a diode-resistor-loop, so you could use a diode as a battery; the reason is that by connecting the diode you get two additional junctions (between each diode side and the multimeter or resistor). at these junctions you get additional potentials which make the total potential from one side to other (these are 3 junctions) being 0V;
to summarize:
the general reason is: perpetum mobile is not possible
the detailed reason is: additional potential differences cancel the pn-junction voltage.