How to solve for current given resistor value for LED circuit?
Is there an easier way to solve for or at least nicely estimate what \$I_F\$ ... because I need to make sure I don't fry my microcontroller!


Two easy solutions:


Choose a target current that's well below the limits of the micro or LED, like 5 mA instead of 20 mA, then just assume \$V_F\$ is close to its nominal value (say, 2.2 V in this example). If the LED doesn't have to be visible in very bright light, 5 mA should be adequate (or you should be able to find a different LED that is adequately visible with 5 mA)


or


Use a transistor buffer with much more than 20 mA capability to eliminate risk to the micro. 


Before you try to calculate the diode current any more precisely, you should consider that the diode forward voltage is likely to vary by 100 mV or more as its temperature changes.