N.
It sounds to me like your husband is jealous and insecure. There are obvious trust issues if he thinks you can't be nice to this guy without it being some sort of validation that what the neighbor said was in some way acceptable. Unfortunately, if you want your marriage to work, sometimes you have to do uncomfortable things to make your partner feel more secure. In this case, what would it hurt for you to stop being friendly with the neighbor? If the neighbor makes a comment about it, you could just say something like "Nothing personal, I just don't want to give you the wrong idea based on your previous comment and I think it's just best we keep our distance." I wouldn't push it off on your husband and I wouldn't be rude. Just stop the pleasantries and friendliness and hopefully it will help your husband feel better about the situation.
I can empathize with your husband in that if I knew a neighbor had an attraction towards my husband, I wouldn't want them to be around my husband very much and I certainly wouldn't want my husband to be all nicey nicey to them. I would much rather offend a neighbor than make my husband feel uncomfortable when I'm trying to repair my relationship and I would expect my husband to do the same thing. That's just me though.
I hope you can find a solution and that you can salvage your marriage as well.
Blessings,
N.