I haven't tried it, so this is just a guess.
How long do you want to run the well pump off of the battery? If it is just a few start/run/stop cycles over the course of a day, maybe you could do it. If you are looking for more frequent usage and/or longer duration usage, then a single battery (without any means of recharge) is unlikely to be satisfactory.
My own well pump requires something like 5,000 watts to start it, and maybe less to keep it running after it first starts.
To do that off of a 12-volt battery and an inverter is not an easy thing. First, an inverter capable of supplying 5,000 watts (and maybe more) at 110 volts is a pretty big puppy - you can get them for maybe $400 or more depending on quality, so it is at least feasible to do from the perspective of finding an inverter.
If the inverter were 100% efficient (which it is not), then your 12-volt battery would need to supply on the order of 400 amperes of current for long enough to get the pump started, then some lesser value while it runs until it is ready to cut off. A typical automotive starter battery is designed to provide such an amount of current for a short period, but it is not designed to provide sustained power at high currents. A deep-cycle marine battery is designed for sustained discharge, but not for super-high current draw. You would have to do some calculations based on the ampere-hour rating of whatever battery you chose (and the discharge curve for that battery) to be sure.
So I'm not optimistic that you will get good results from a single battery. Maybe a battery bank would be better, but even with that my best guess is that you are looking at a short-term proposition unless you have some method of recharging it.