I can't imagine the old one damaging the battery. That's not to say the battery doesn't have an issue, but if it does, it shouldn't be from the old alternator. It would have to be overcharging like crazy to do that.
A fully charged battery with the surface charge burned off should be 12.6 volts. So if you charge it, then burn the surface charge off by cranking the engine for a few seconds (but don't start it...kill the ignition) it should read around 12.6 volts. There are cycle tests, load tests, and others that can be done with some good equipment, but if you pass the voltage test and it isn't going dead, etc, I'm betting it's fine.