Breakers (at least the traditional kind) respond only to current, not to voltage. If you don't have any other 120V appliances that are damaged, you are quite fortunate indeed. Many smaller electronic devices these days will work fine on either 120 or 240 volts (or most anything in between), but larger appliances like microwaves and air conditioners and so forth are usually voltage sensitive.
The converter charges the battery when on shore power.
It's unlikely that there's a fuse between the converter and the 120V distribution panel. It's fairly common for it to be on its own breaker, so checking that all the breakers are not popped is a good idea. Some breaker designs will look like they're on when they've popped.
There may be an internal fuse on the converter itself. If there is and it's blown, the converter may still be blown. The purpose of any fuse like that is primarily to prevent a fire and only secondarily to prevent damage to the converter.
Testing the converter is pretty simple. If you put 120VAC in and you get nominal 12VDC out, it's working. If you get something else out, like 0V or 15V or 23V, it's not working. If there are sparks flames, it's definitely not working!
If you do need to replace the converter, I'd suggest thinking about an upgrade. The Progressive Dynamics 9200 series consistently gets very high praise here, though there are others that are also well regarded.