Well if there is no battery, the circuit is pretty much useless... The p-n junction is typically associated with a voltage drop across the p-n junction (somewhere around .5-3V for LEDs). However with no battery in the circuit, there will be no current flowing and thus no voltage across the junction, so they should be at the same potential.
Is the battery attached to the n-side? If so, then yes, the n-side would be at a higher potential and the p-side would be a 0V, since there in no (practically none) current flowing through the diode. This is known as the reverse-bias direction. The voltage at the n-side would be at the same voltage as the battery (assuming it is just connected with a wire).
Look up "PN junction depletion region". The natural potential that develops spontaneously across a PN junction is due to the diffusion of charge carriers across the junction. P type material has "holes" in its bond structure that can be "filled" by accepting electrons, while the N side has more than enough conduction electrons to satisfy its bond requirements.
However, every electron that crosses over leaves a small net positive charge in the N material, while the P side gains a small negative charge. These charges build up until they suppress further movement --- they raise a potential barrier against further diffusion of charges. This is what leads to the "turn on" bias requirement to get current moving across the junction (about 0.7V for silicon PN junctions).