@Josie86 Sounds like this guy is a total scammer, just looking for ways to get a refund. It's too bad you offered him anything, that really wasn't necessary and sets a bad precedent for him thinking he can pull this sort of thing on hosts. Tell him that offering him half refund was incredibly generous of you considering he couldn't show you where this happened (which is quite suspect- if he hurt his foot that badly in your home, he'd be able to identify approximately where it happened, and there would be some nail area where the paint had been damaged). Then contact Airbnb and tell them the scenario and that you feel you are being scammed for a free stay. He is bound to contact them demanding a full refund, so best you get your foot in the door first.
Whatever you do, even if you gave him a full refund (don't, don't, don't, I wouldn't even give him anything) , he's the sort who will leave a bad review, so don't be thinking you can ward it off by being accommodating.
Make sure all communication with this guy stays on the Airbnb messaging, not verbal or text, so Airbnb can se a record of the interactions. If something is verbal, flllow it up with a message-" So, XX, just to clarify, in our verbal conversation yesterday, you indicated that you wouldn't leave a bad review if we gave you a full refund for the small nick on your foot that you claim happened in our home on a nail, even though you couldn't show us where that nail was?" Or whatever the verbal exchange was. The best scenario is if he does communicate in writing that he will leave a good review if you refund him- that's extortion and totally against Airbnb TOS.