It would seem to me by his denial that he probably thought nobody knew it was him shooting the birds.
I would hope that you challenging him has put the frighteners on him and it won't happen again.
Perhaps obtaining one of the pamphlets that explains the GL and airgun law and putting it through his door might help educate him especially if he's an older man who simply might not be aware the law has changed dramatically from his childhood. (unlikely unless he's just came back to airguns after decades away)
Either way, if it happens again the police absolutely do need to be involved.
Im sure your wife will make it known among the neighbours what happened in your own garden,, just in case some of them know you have a rifle and thought perhaps it was you shooting the birds.