Will it damage the battery?
If the power adapter (that's the proper name for a "charger") survives and keeps delivering current while the battery voltage increases": Then yes, the battery voltage will increase too much, electrolysis will take place, hydrogen will be formed which can explode when mixed with air. Your battery will not be the same as it was. So yeah, it will be damaged.
The power adapter might also be overloaded, best case, it will blow an (internal) fuse so then you cannot use it anymore either. I think this is best case because this will also stop charging the battery so win-win.
Worst case, it might overheat and catch fire, then you also cannot use it anymore.
Summary: just don't. Buy a proper Lead-Acid battery charger. They're not that expensive.