Car batteries DO explode. The acid/electrolyte itself isn't flammable.... what is flammable is the hydrogen gas that is being produced as a by-product of a lead-acid battery's chemical reaction.

Under the right conditions, subjecting a sufficient concentration of hydrogen+air to an ignition source, and it will go kaboom!

Hydrogen collects at the top of each cell compartment so when they ignite, the sudden increase in pressure cause the top of the battery to rupture. Those are typically the weakest points of the battery. Just do a google image search of exploding car batteries.