Voltage drop on diode
You can measure diode current right down to zero volts. However, below the magic figure of 0.7v (for silicon), it tends to be very small.

It depends on the type of diode what the current is. Here is a picture of some measurements I did a while ago to answer a related question. Do click on it if you want to read it. What surprised me is how bad the 3v zener was at stabilising a voltage.



So how do you figure out the diode voltage?

For a first order approximation, we know the diode current is going to be 'very small' (as the voltage is below 0.7v), so we can neglect the resistor voltage drop, and say the diode voltage is about 0.5v.

For a second order approximation, you can see from the figures above the current for my 1N4148 at around 0.5v at whatever ambient temperature it happened to be was about 100uA. That dropped over 100 ohms is 10mV. So with more accuracy we can estimate the diode voltage as 490mV. Interestingly, the figure for a much bigger 1N4006 turns out to be very similar.

There's absolutely no point going to a higher order approximation without specifying the diode, and the temperature.