Using only a simple resistor and diode won't work well as the resistor will generate quite a lot of heat.
lets say for example, you're using 120VAC mains, that same 2v Red LED, a silicon rectifier diode, and a resistor.
And wanting to drive it with ~10mA.
Then 120V - 2v(Led) - 0.6v(Diode) = 117.4v across the resistor.
117.4v / 0.010A = 11,740Ω.
117.4v * 0.010A = 1.174 Watts.
The resistor will generate more than a watt of heat....just to power a small indicator LED.
A better solution is to use a capacitor's reactance to limit the current.
Some topics like
this or
this could provide some insight.
In my opinion, if the neon indicator is still fully functional, just keep it as it is since it only requires a single resistor to function properly and unlike the LED, it can work on AC and DC current.
Neon lamps even come in different colors like LEDs do...just not as many.