Most men enlisted in the county ar at least the state he lived. He was born in Brooklyn NY and was living in Nassau, NY when filed WWII Draft Registration Card in 1942 stated "unemployed" and a "disabled war veteran". So try the NY State Archives and the county library where he lived. The local newspapers of the time had more local stories of the comings and goings of the local folk. There might be a mention or listing of men enlisted or units served and where going.
Another thing I have found since Google Books: search his name there and see if he is mentioned in any books published about the war either research ones or memoirs of other soldiers. Search full, then Charles and Charlie.