The various branches of the military switched over from using service numbers (issued serial numbers) to using Social Security Numbers (SSN) for identification purposes some time between 1969 and 1973, depending on the branch of service.
The SSN was even a part of the service member's mailing address and was written on the envelope of all mail to or from.
Prior to the idea of keeping SSN's protected information, they were included on all official listing such as unit rosters, reports, flight manifests, awards, etc. Many of those records are now on microfiche in the military records of the National Archives II, College Park, Maryland.
If you know this Marine's units and dates of service, it might be possible to locate him on a day report or unit roster, along with his rank and SSN.
Due to restrictions imposed by the Privacy Act, US Government officials are prevented from giving out SSN's, but the numbers are in the records and available to any person researching those records.