Sharon,
Chaim Greenberg was 82 or 83 when he died in 1953, so he was born around 1870. That makes him a bit too young to have served in the US Civil War. His death certificate states that he served in the Spanish American war which was in 1898, when he was in his late 20s.
The inscription on his stone "Erected by East Side Post J.W.V. #4" refers to the "Jewish War Veterans of the United States of America". While Post #4 may be defunct, the organization still exists. See: https://www.jwv.org/

I hope this helps,
Rodney Eisfelder
Melbourne, Australia