It would require some further diagnosis. Basically, that light is lit by battery voltage. Normally, the alternator charges the battery in the other direction, thereby cancelling out the 12V battery light. If the light stays on, that usually means it's not getting enough voltage from the alternator (13.7 to 14.5, something like that) to defeat the light.
Since the light does go out eventually, we might suspect a lazy alternator or defective voltage regulation. I don't think it's your battery. However, corroded battery terminals could possibly cause this problem.