Does a blown fuse cause the battery warning light to come on?

Yes. You're getting low charge from the alternator. An easy to test would be to replace that bad fuse and run the as many accessories as you can for a while. If the motor dies in about ten to fifteen minutes Then you know it's the alternator. Or you could just put a volt meter on it and measure the charge output. Anything less than 13V DC means you need a new one.  more

I would test the alternator first. Start your car then unhook the positive wire from the battery. If the car stays running after a couple of minutes its not the alternator. Then I would look for a damaged wire or a place where power could be arcing of a wire somewhere. As this could lead to a power drain and cause the light to come on.  more