Hi Elijah,
I’d suggest a couple of things.
The verses concerned are 1 Cor 11:4-5: “Any man who prays or prophesies with his head covered dishonors his head, but any woman who prays or prophesies with her head unveiled dishonors her head – it is the same as if her head were shaven.”
A zucchetto does not, in fact, cover one’s head. The covering suggested here by St Paul is something more like a complete head covering, as in a hijab. So, I think your friend is mistaken here.
Secondly, is he then advocating that all women at church (since they are praying) should have their heads covered?
God bless.
Jonathan