Okay, as a celebrity his genealogy is pretty public. His immigrant great-grandfather was Giuseppe Fieri. It’s a little harder to track down where he’s from, but the one tree I found on Ancestry said he was from Arsago Seprio in Lombardy. If that’s the case, his pronunciation seems off, but I’m a little dubious about the source, as Fieri isn’t really a Lombard name. It seems to be from Tuscany and Lazio.
I’m still inclined to think the explanation given above is correct, that he heard /fiεɾi/ and was used to only hearing /ɾ/ in words like wedding, and maybe was able to contrast it with full-on rolled /r/, so mis-parsed the R sound in “Fieri” as a D sound.