The LM393 comparator used is not really designed to drive LEDs. The current limiting circuit causes a voltage drop at the output so you should NOT try to pass any more than the minimum spec of 6mA.
The output current limiting is severe and may result in destruction of the device.
Read Note 7 in the datasheet and understand that at just 20mA the device dissipation could rise to beyond 2W ....clearly far beyond its rating. You can understand the rather primitive current limiting by examining the schematic diagram:
The output transistor is driven by a constant current source, so depending on hfe the device rapidly pulls out of saturation at some given point not related to the supply voltage. While the datasheet specifies the output can be shorted to ground continuously ....you'd expect that for an open collector output!!! What they say about a short to a positive supply is that you MAY DESTROY THE DEVICE. For this configuration with +/-12V supplies YOU CANNOT short the output to either ground or +12V.
So @Dave Tweeds comment about short circuit capability of the device is completely wrong.
In this particular application with +/-12V supplies I would suggest that a short to ground or a short to +12V would instantly destroy the device.
There are several ways you might alter your circuit to provide more LED current in a reliable fashion. Given that you have already chosen to use +/-12V, I'll suggest only modifications to that schema (it would not be my first choice):
simulate this circuit – Schematic created using CircuitLab
The LED is driven by a constant current source (Q1, 2N3906), so the output is short circuit to ground friendly. This is important if your LED is panel mounted and you have to run wires to it (You should never run your supply voltages in external wiring without current limits in place). In fact the collector output of Q1 could be shorted to +/-12V or ground without damage to the device.
How does this work?
Q1 works as a constant current source limiting the current through the LED.
The advantage to setting the LED current with a constant current source is that you now don't have to consider the LED Vf. You could use any LED.
R4,5 limit the current sunk by the window comparator to approximately 4mA. My original calculation set the divider current to 10x the base current required, but that was just to be ultra conservative. R4,5 provide a fixed base voltage to Q1 of about 8 V.
You could reduce the current through R4,6 further if you want to. For example you could raise R4 to 2k and R5 to 10k. This would drop the comparator current to about 2 mA which is still enough to define the base voltage for Q1. The base current for Q1 is likely around 400 uA so any resistor selection has to take this into account.
You are of course limited in how high you can set the LED current. You are limited by the power dissipation in Q1. In the circuit shown with just a single LED load (Vf=2.2 V) you may see Q1 dissipating a maximum of about 130 mW. Well within rating, but it will get warm to the touch. With a Vf=3.2 V this power dissipation is about 110 mW. You could easily double the LED current to 40 mA and still be well within the rating of the 2N3906.