Most "acid" you get when buying a battery is already mixed with water. Regardless, it should never be added to a functioning battery (even a moderately discharged battery) in place of distilled water. Never use tap water or de-ionized water to fill a battery.
For severely discharged wet-cell batteries that won't come around after water top-off and extended low-amperage or pulse charging, you can dump the old acid out and try adding new, yes. Chances are this won't bring the battery back to life for more than a short period, but sometimes that's enough. If it won't hold a charge though, it's toast and time to buy new.
Additional acid or water should never be added to a maintenance-free (absorbed glass mat or AGM) battery, regardless of their status. The seal-strip on these batteries is there for a reason. More water or acid can cause AGM batteries to swell and crack, so once they're filled and fully charged, that's it.