How do I determine the output voltage in this diode circuit schematic?
The +15V is a red herring and completely irrelevant since the current source delivers 1mA of current, no matter what it is connected to.  So you have to add up your voltages from the only potentials relevant to this circuit, which are the cathodes of the diodes: the diode with the lower cathode potential will be conducting and the resistor will add a fixed voltage due to it being driven with a fixed current.