This does not solve the mystery of the second American in the photograph, but I can provide information about your grandfather and when he likely was in Egypt when the photograph you shared was taken.
I have found a Robert J. Martin, Jr. aboard a Liberty ship named WILLIAM H. WELCH on a voyage that included a port call in Alexandria, Egypt, in October 1943. Robert J. Martin, an ensign at the time, was in command of a 26-man Armed Guard detachment on the ship. His serial number appears in the record and it matches the number you have, 166616. It appears that WILLIAM H. WELCH departed New York on or about May 1, 1943, in convoy NK-538, destination Key West. I cannot trace the ship until finding her much later in Basra, Iraq, departing there on or about September 7, 1943. (Obviously she had to have crossed the Atlantic and the Mediterranean, then the Suez Canal, the Red Sea, the Arabian Sea, and the Persian Gulf.) Subsequent to her appearance in Basra, the ship was in Bandar Abbas, Iran, then proceeded to Aden, Yemen. From there she presumably proceeded back through the Red Sea and the Suez Canal to Alexandria. On October 19, 1943, the ship joined convoy GUS-19 that departed Alexandria on that date, proceeded west through the Mediterranean and then the Atlantic, finally arriving Hampton Roads (i.e., Norfolk area), Virginia, on November 15, 1943. One can assume the photograph dates from early or mid-October 1943.
I also find your grandfather on two consecutive voyages aboard a tanker named W.W. BRUCE, the first from New York to Curacao, to Londonderry, Northern Ireland, and returning to New York in October -November 1942. The second voyage was from New York to Glasgow, Scotland, returning again to New York between November 1942 and February 1943.
It is very likely that he was assigned to additional ships (you mention he was in Okinawa) but I was unable to find other records. It is possible that he is listed in my sources simply as Robert Martin but that is much too common a name to find him easily.
There is a tragic postscript to your grandfather’s time aboard WILLIAM H. WELCH, which apparently did not involve him. On her very next voyage, WILLIAM H. WELCH departed New York on December 22, 1943, destination Liverpool. It was typical for all or nearly all of the merchant marine crew aboard a given ship to be replaced at the end of a voyage, and likewise most or all of the Armed Guard detachment would be replaced after having served on a ship for several voyages and/or several months. Therefore, I presume your grandfather was no longer aboard WILLIAM H. WELCH when this later voyage commenced. After delivering her cargo in Britain, WILLIAM H. WELCH began her voyage back to New York. However, on February 26, 1944, in a fierce storm, she was driven ashore near a desolate island in northwest Scotland where she broke apart. Out of a crew of 74 merchant seamen and Armed Guard sailors, 61 were lost. There is a detailed description of the incident, excerpted from the book “A Cold Corner of Hell: The Story of The Murmansk Convoys, 1941-45,” by Robert Carse, at http://www.armed-guard.com/ag88.html.
I am guessing you or your family do not have your grandfather’s military records, in which case you may be able to obtain a copy of his official military service record. Please see this page from the Armed Guard website that I manage: http://armed-guard.com/searchmil.html. In particular see section A.1. Records of Individuals - U.S. Military. You will have to contact the National Personnel Records Center in St. Louis, MO, a facility operated by the U.S. Archives. Provide as much identifying information as possible about your grandfather; his ID number will be a particularly critical piece of information, possibly second only to his name in importance. The links on the web page above will take you to the necessary pages of the Records Center web site. There will be a fee for researching, photocopying and mailing the information but the Records Center staff will not begin research without informing you of any charge. His military service record would contain such information as dates of enlistment and discharge, the ships to which he was assigned, applicable dates, training, any decorations or medals earned, etc.
Note that someone such as yourself, who is not next of kin to your grandfather, may not be able to obtain his full service record (next of kin = parent, spouse, sibling, child). If there is someone still living in your family who is next of kin to your grandfather, such as your father or an uncle, it would be best if the request to the Records Center was submitted by that person. You can do the legwork of research, completing forms and otherwise preparing the request but the request should be signed by that person. If there is no longer a person who is next of kin still living, then make the request yourself and hope for the best. There should be something of interest to you even if not his full record.
Good luck and best wishes.
Ron Carlson, Webmaster
Armed Guard / Merchant Marine website