I did try giving it some gas while starting up and it didn't help. I didn't have it floored the entire time, but I did try it quite a few times, with variations on how far I pressed the pedal.
From what I can tell between what the service guy told me and what I've learned online, the battery was probably the culprit for my car not starting up. The battery is about 4-5 years old now, and I'm sure the 12" subs in the trunk haven't helped either.
Since the dealership found water in the gas, I'm guessing I got lucky the battery failed when it did, since it allowed me to catch the fuel problem as well, before that started doing some serious damage.