Yep, everyone else said it. The only way to get a current different from the load into the regulator is with a parallel connection. To get what you were thinking of by using the voltage drop of the regulator to light it, you would hook it in parallel with the regulator in and out. However, I have no idea how this would affect the regulation.
Hooking in parallel with the input, you just need a resistor to consume the waste voltage at 20 mA. 14V - VfLED = 10.3V in this case. V = IR, so R = V/I or 10.3V/0.02A = 515 Ohms. Common value at 510 Ohms should be good.
Now, make sure the power is good. P = IV = 10.3 V * 0.02A = 0.206W 1/4W will be fine, although close to max rating.
I over voltaged a 3mm LED and it had an air bubble around the chip. When it got really hot, really fast, it blew apart and dented my ceiling. Scared the crap out of me.
Edit: Just saw your response. Look at it this way:
If you remove the LED from your schematic (replace it with a wire) and hook up a load to the 9V output that pulls 1 A. How much current needs to flow through the wire that replaced the LED? A little over 1 A, due to losses in the regulator. If you put anything in there, either it has to handle 1A or limit the output. Nothing else is possible.
If the electrons flowing were people (coulombs of charge) walking down a hallway. They are walking at a certain speed (amperage), due to circumstances you can't control. Now you get out there in your crutches and can only walk 1/50th of the speed of them (1A vs 20mA). What happens? You get blown up! Slammed to the ground.
Now if you have a side hallway that is narrower (current limit resistor) that runs in parallel with the high current hallway. The both start and end at the same place, but you can travel slower in the small hall without getting blown up. That is what hooking the LED in parallel does.
The voltage is what is required to power the LED if nothing else affects it. What your circuit was doing is providing the voltage, while also forcing current through it. Both are important. What probably happened is the voltage across the LED was WAY over 3.7V. Because current needed to flow, so the only option was to power through the increasing load of the failing LED, until it blew.
Take a look at Forward Voltage vs Current on page three of this data sheet (probably fairly similar to the LED you are using):
http://www.vishay.com/docs/81260/vlww9900.pdfThere is a Max value of Forward Voltage. After this, so much current flows through the junction that it melts. Boom. You get what you got. The is nothing in an LED that will limit the voltage to a safe value. This is why you need current limiting resistors. There is a linear relationship between resistance and current for a given voltage. You can see in the Forward Voltage vs Current that this doesn't exist for an LED. After it turns on, the current increases MUCH faster than voltage, so it can't limit itself and will explode with too much voltage.