Determine supply voltage for led from current limiting resistor
Sorry, but no. Attempting to drive an LED with a constant voltage is a great way to kill an LED. The coefficient of forward voltage with temperature is negative, which means that, driven by a constant voltage, if the LED starts to get warm it will start to draw more current, which will increase the power dissipated by the LED, which will cause the forward voltage to drop even more, etc to disaster.

You are better off assuming an LED forward voltage of about 2 volts, then provide a convenient voltage/resistor combination to provide ~ 1 mA, then measure Vf. Then reduce the resistance so as to produce a series of currents in the 1-10 mA range, and at each current setting check the encoder for proper operation.