Can the potential barrier across a p-n junction be measured by simply connecting a voltmeter across the junction?

No, we cannot measure the potential barrier across the p-n junction by a voltmeter as this potential is only in the depletion region due to the electric field in that region. The rest of the crystals are insulator and thus block this electric field and as the voltmeter is connected across the diode, the barrier potential would not be measured.