Understanding the PN-Junction Built in Voltage
How can a charge neutral substance have a potential?


To say that a PN junction has built-in potential isn't to say that the PN junction has a potential relative to ground or infinity etc.


  How can there be an inherent potential in a doped semi-conductor if it
  is charge neutral?


Charge has been separated within the PN junction and, thus, there is an electric field across the depletion region and an associated potential difference.  A charged capacitor is neutral but there is a potential difference (voltage) across the dielectric.  There are some similarities but...


  However, I don't understand why we can't measure the voltage drop
  across the PN block with a voltmeter.


As explained, for example, here, the built-in potential is not readily measured with, e.g., a voltmeter.  In other words, this question has been asked here several times (which means you are not the only one perplexed by this - most are at first) and there are good answers already available.