It should, but it might be one of those things where if the battery still has voltage the battery light does come on, but the ECU still gets confused by the drop in voltage?
I'm trying to remember what happened when mine died, and I'm pretty sure the <check engine> light came on first... then the battery one... then it died completely. - That was an accelerated process though because it was night and it was also cold and raining; so lights, fan, and wipers all going at the same time.