I wouldn't worry about social security numbers or service numbers. The former didn't exist prior to the 1930s, and my understanding is that the Navy didn't use service numbers until 1920.
Could you please share his name, birth date, and place of birth? Depending on exactly when he served it may be possible to look up additional information on Ancestry or a few other resources.
Dear Mr. Meyer,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the Official Military Personnel Files, 1885 - 1998 in the Records of the Bureau of Naval Personnel (Record Group 24) that may include a file unit for your great grandfather. Only a few of these of these file units have been digitized. To request a copy of the non-digitized records, please mail a completed GSA Standard Form 180 to NARA's National Personnel Records Center (NPRC), (Military Personnel Records), 1 Archives Drive, St. Louis, MO 63138-1002 and attach a copy of any documentation you may have such as his registration card.
Due to the coronavirus public health emergency, the NPRC closed again as of November 7, 2020 until further notice. NPRC will respond only to requests involving burials, medical emergencies, and homeless veterans. If your request is urgent, please see Emergency Requests and Deadlines or emergency requests may be faxed to (314) 801-0764. Please check archives.gov/veterans for updates to the NPRC operating hours and status. We apologize for any inconvenience.
We hope this is helpful. Best of luck with your family research!