Water has already been burned. It’s a combustion product. And as such, it has no energy to give. The battery may need water to operate (like maybe as an electrolyte). But water has no energy to supply. In a way, it’s kind of like those potato batteries common in grade school science projects. Although potato starch actually COULD supply energy (with air as an oxidizer) in a fuel cell, in these batteries, it’s actually the metals that supply the energy.
Very true about water made of antimatter. Of course, even if we’re talking about plain old ordinary water, you could get SOME energy by oxidizing it with something more powerful than oxygen (like fluorine). However, fluorine would be required as an oxidizer. And this would certainly not be very ‘green’, especially considering the toxic nature of both fluorine and the resulting hydrogen fluoride.
I came across an old technical paper on a proposed rocket engine that ran on fluorine and powdered uranium. Also one that used mercury oxide and powdered uranium. Damn EPA would probably put the kibosh on such fun these days…