The very short answer is yes. But, it's a bit more complicated.
The ability of your inverter to power power more than your fridge depends on the number of watts the current inverter is capable of supplying minus the power required to run the fridge. Not likely that your inverter much bigger than what's needed to run that fridge. In addition, unless you have added additional batteries, your ability to supply power to that inverter is also limited. So it really depends on what AC appliances you want to operate (watts) and for how long.
By way of example, I have 4 solar panels on the roof, two 175 Amp Hour Lithium batteries and a 3 kilowatt inverter. Do not have a residential fridge however. With that setup, I have been out boondocking for 10 days at a time with no problems at all. In fact, I hardly ever run the generator any more in that situation.
As far as the wiring....It can be as simple as adding a additional single outlet to the existing fridge circuit, or as complicated as having a
(much larger) inverter run the whole RV. In either case, these changes should only be done by someone who is very familiar with electrical circuits and qualified install them properly.
Hope that helps.