I hope you’re feeling better about using the name Emilio. It’s a cool name, I especially love O ending names!
You’ve been given so many arguments here, including that Emilio is a Latin (Italian, French, Spanish & Portuguese) name. Moreover, Western Europe is historically very intermingled (the Romans conquered the UK for goodness’ sakes!).
Also, as above, does your friend have the same feelings about the top Italian names of recent decades including so many “non-Italian” names? Is he worried about William, Catherine & their three children having Germanic, Greek & French names?
If Emilio is suspect then so are most names used in multicultural countries today. Goodbye Leo & Sophia, just to name two!
Cultural appropriation is when you take from a country or group that has been historically marginalised and exploit the benefits but share none of the burdens of that culture, or you are somehow making fun of/ mocking that name/practice/custom. This is clearly not the case!
Also, it sounds like you friend lives in the UK - and therefore supports multiculturalism and cultural exchange? Maybe he’s feeling especially homesick right now or some intense defensiveness about Italy for some reason. This might make him feel like you’re taking something you shouldn’t. But this is his personal feeling, this isn’t shared reality.
I would go for the open but firm chat, give him some counter arguments, and ask him if there’s anything else going on. But make it clear you’re using the name, hope he can soften and get on board but will be going ahead with it. Don’t change your plans for a friend (or anyone).