Can voltage drop across a diode be controlled by varying the doping level?
Without going into solid state physics more than I am competent to : no.
You might be able to shift the bias by a few millivolts this way by changing doping, but you can do that more easily by changing the temperature or the bias current. Read up about the Shockley equation - you can use a diode to generate logarithms that way...

To get a bigger change you need to move to a different semiconductor material (germanium for about 0.2-0.3V, silicon carbide for higher voltages (dim memory says about a volt...) anyone know the numbers for 3-5 compounds like Ga-As, Ga-InP etc?

Finally, Schottky diodes involve a metal-semiconductor boundary and have quite a low bias voltage (about 0.3V)