Seeking grandfather's birth certificate

If he was married in a Roman Catholic Church, sometimes the church marriage record contains the name of the church where the person was baptized, as proof that they were Catholic.  I have traced a few people back to their birthplace using this method, but other times the information is not part of the marriage record.

In my area, the local Catholic parishes have turned their records over to the diocese.  You might try that.

A few places you could try:

ItalianGen.org (free) has a couple of birth databases as well as a baptism database.  (It is an overall database, not just Italians.)  This may help if his birth was registered in a county other than Nassau.

https://italiangen.org/databases/

ReclaimTheRecords has birth indexes on Internet Archive (free).  In the search box on the left, type in New York Birth and it will bring up all of the possible record sets.  There is a mixture of state and city records.  You can limit by year on the left.  Make sure to check the years around 1881, as well.

https://archive.org/details/reclaimtherecords

FindMyPast.com (pay site) has New York Roman Catholic records, as well as some other cities.  There are baptisms as well as marriage records. I am not sure about the parishes covered.  Susannah’s comment about checking the marriage entry (if he married in the Roman Catholic church) is a good one.  I, too, have seen mention of where a person was baptized in the margins of the marriage record.  FindMyPast updates these records periodically so check back if you don’t find the one you need.

https://search.findmypast.com/search-world-Records/new-york-roman-catholic-parish-baptisms

Newspapers - it may not be likely, but I have seen mention of christenings in a newspaper.  It is more likely if the family was middle class or above.  You could check Chronicling America, Newspapers.com, NewspaperArchive, GenealogyBank, etc.

Happy Hunting!