Firstly, let me offer my condolences to the family of the Irish citizen who died in New York.
Officials at the Consulate General in New York have advised that the process for obtaining a death certificate depends on whether the person died in New York City or elsewhere in New York state.
In the case of deaths in New York City, full details of how to obtain a death certificate are available on the website of the New York City Department of Health and Mental Hygiene at http://www.nyc.gov/html/doh/html/services/vr-death-general.shtml.
In the case of citizens who die elsewhere in New York State, a death certificate can be obtained from the New York State Department of Health. Full details are available on the New York State Department of Health website at https://www.health.ny.gov/vital_records/death.htm.
Alternatively, I am informed that there is at least one private company authorised by the State/City of New York, who will process an application for a death certificate online and send the certificate overseas. Further details for such services are also available online.