A green LED takes slightly more forward voltage than a red one. Probably in the neighborhood of 1.8 V or so.
An LED will not put out any light unless you apply enough voltage to forward bias the junction. A single 1.5V dry cell will not be able to light a green LED.
An LED also needs a series resistor to limit the forward current. If you exceed the forward current rating, the LED will quickly overheat and die.
