Tony:

I am switiching this conversation to the Research list since this is really a research topic.

There are 26 Gerardo Diaz in the Social Security death index. Where did your gradfather initially live/work when he immigrated to the United States? Where did he cross the border (El Paso? Nogales?) Do you know when and where he died?

If you search under Gerardo Diaz, there is only one record for someone whose SSN was issued in Illinois a and that person was born in 1927 so I doubt it is your grandfather... so I tried searching under Jose Diaz... since in Mexico people's official church names were often preceding by Jose or Maria even if they never used that part of the name in real life... i.e. someone named Marcelo Garcia would be listed in their baptism record as Jose Marcelo Garcia...

Under Jose Garcia there is this possible lead:

JOSE DIAZ 19 Nov 1892 Feb 1976 83 Chicago, Cook, IL 60625 (not specified) Illinois 345-46-0754

You would have to confirm that your father first worked in Illinois (and therefore applied for his SSN there) and his death date. If this is him, you could simply order his application for an SSN from the social security administration using the SSN listed here.