Your questions tend to have no wrong answer as both YES and NO applies.
Can You charge them by running the generator YES but it will take hours.. How many depends a bit on how far down the batteries are. but a full recharge from 50% SOC (Deep cycle only) is about a six hour job. From 75%SOC (MARINE/deep cycle) one hour less. from 90% SOC. about 2 hours. (both types) this assumes the proper (C/30) size converter)
Does the grenerator charge the batteries NO. not directly but it powers the converter and THAT charges the batteries so the answer is yes.
See what I mean by no wrong answer.