Your history is a lot like mine... My GG-Grandfather born 1857 in Salerno, left Italy ~1883, G-Grandfather born in Brooklyn 1887, Grandfather born in Bronx in 1919.
I will caution you that you are going to be needing _a lot_ of documents and you'll possibly run into issues like I did: My gg-grandparents had 7 kids in Brooklyn. They only reported the last child in 1897 to the city/state. I have baptisms for 4 of the 6 "unreported" births, but the city has no records.
Based on
http://cognomi.sapere.alice.it/it_map.php and looking up both "Davide" and "Cordialino" I would believe that "Davide" is the correct surname from the Campania region, since there are no "Cordialino" surnames that turn up for Italy. Those records are current, but the lack of any names gives more confidence.
I trust you have seen
http://italiangen.org (or Steve Morse as vj suggested) and used their vital record database?
Surname: Davide
Name: Cordialivio
Age: 62 y
Death: May 23 1922
Cert #:11494
County: Kings
If you don't mind spending the $10 you could request the cert online at:
http://www.nyc.gov/html/records/html/vi ... eath.shtml
For reference, I requested ~20 death/marriage/birth certs from NYC.gov and almost all came within 3-4 weeks (mid-July 2008).
I ended up requesting any cert for the immediate family. It ended up costing a fair bit, but it helped me find new spellings and I made connections, such as them all living at the same house for decades.
Lastly, I had contact with the Archivio di Stato di Napoli and they don't have their military enlistments digitized and they have incomplete records from 1842-1930, so they can't look up by name to find comune. One technique is to email the state archives and see if they can find the person based on birth year, but it doesn't look like Napoli can help you based on my email from them on August 12th. Salerno on the other hand was able to find my ancestor without comune, but they had him listed by his 2nd of two middle names.
If you can lock-down the year with high confidence then you can give the Archivio di Stato di Napoli a shot. They took about a week to respond to my email, which was in my broken Italian.
I dealt a lot with NYC/Brooklyn so I may be able to help with questions around that process.