Dr. Ergun Caner's Father legally changed his name from Acar Martin Caner to Acar Mehmet Caner (link to evidence provided courtesy of Jason Smathers).
I suspect that this was not the first name change in Acar's life, since "Martin" sounds like a nod to Acar's wife's Swedish Lutheran heritage. Likewise, I cannot help but wonder if the legal name change back to something more Turkish-sounding was motivated by Acar's second wife, who was apparently Turkish. These suspicions are just that - suspicions.
What the evidence does suggest, however, is that while Acar may have a devout Muslim to some extent (and perhaps increasingly so in the late 70's and early 80's), at some point he seems to have gone along with using (as his legal name) the rather "Christian" name Martin. That does not sound like the sort of thing that an extremely and zealously devout Muslim would do, particularly if that "Martin" were replacing the name "Mehmet" (a Turkish form of the name Mohamed).
Then again, perhaps there is a way that someone's name could be changed against his will in the immigration process - or something like that. This evidence is just evidence, it is not proof.