Many inverters do include not only an inverter but also a battery charger and a transfer switch all in the same case. These are more properly called inverter/chargers but most people just refer to them as inverters even though they do all 3 functions.
OP, I don't know about your particular unit but in general the answer is "maybe". You need to have the charging function switched on. You also need to have it programmed appropriately for your particular batteries. This could be done with jumpers, DIP switches, or actual programming depending on what you have. Better models will have a battery temperature sensor to adjust the charge voltages based on temperature. Then, you have to know if your unit is functioning properly or not.
A quality, well functioning inverter/charger will take excellent care of batteries long term.