There will be a fuse from the battery
to the inverter, but there will (should) also be a fuse protecting the AC load - this is typically on a panel or switch, but may be built into the inverter itself. Here is where one needs to know what type of inverter it is. Not the exact brand, etc, but the power output. If it is a 200 watt pocket inverter, these are pretty harmless (and useless for more than charging electronics
or running very small loads like a TV). However, if it is a 2000W, then that is another ballpark altogether.
The inverter certainly hooks to the batteries. If the old one had a 20 amp fuse on the battery
side, then it must have been a small pocket inverter. If you replace it with a similar one (the important criteria is the wattage size), then there will most likely be no problem.
The fuse has to match the current
draw on the wire, so the inverter has to be fused AND wired sufficiently. You may not be able to drop a larger inverter and fuse on your present wire (hence the need to know the wire size and inverter size before being able to give much advice).
If it is a pocket sized one (or even larger), you can find inexpensive new replacements
at Walmart, Canadian Tire, etc and not have to get a used one from a consignment shop.