Yes, that could definitely be the problem..the green light. However, that little light should take 2-3 days to wear the battery down.
And check the water level in your battery.
When the car is running, put a voltmeter correctly across the + and - terminals. You should get a reading of 13 volts. 14volts is too high. (voltage regulator problems)
If you think your battery is good, turn on the headlights. Start your car, while a friend observes the headlights. If your headlights go dimmer upon starting the car, then the battery is weak to begin with.
I don't know how to tell you to open the dash and get to that green light and its wiring...sorry.