I disagree that the LED replaces 3 components compared with using an LDO. In this case the LDO would share bypassing with the other IC so this solution really only saves a few cents and marginal board space, ie an 0402 LED vs an SOT323 LDO. I would personally take the peace of mind and go with the LDO.
This is a trick that students usually come up with. I may say this is a simple trick but not always be useful. It may only work well if the load is really light and the tempco. (in other words, the output voltage accuracy) is not an issue. To nowadays demanding digital stuff, it surely is not a good idea. However, for many low voltage consumer stuff, it is still a viable solution but need to be used with great care.
I would be concerned about the variation in forward voltage drop across the LED over temperature range (0 to 80 C for a typical consumer device). Typically, core logic operating at 1.8V requires a +/-10% regulated supply voltage.
Given that the 3.6V supply has its own tempco and adding this to the LED tempco, it is possible that the 1.8V supply could have a variation over temperature exceeding 10%.
This assumes the VCC current is relatively fixed to generate a nearly constant forward voltage in the green LED. It also assumes the larger supply is exactly a diode drop greater than the logic supply needs to be. That almost never happens. Interesting trick, but not exactly a universal solution.
t.alex: I'll pass your query to the author. Of course, the circuit is so simple you could try it with and without the resitor, and see for for yourself! What have you got to lose, except an inexpensive LED?