Is it correct to say '...he found out she was sick only when they got married, her turning 18, him 19' or 'she turning 18, he 19'? is the gerund form for the verb 'to turn' preceded by objective pronouns?
Are you trying to say that he was 19 and she was "turning 18" (as in, not quite 18) at the time they got married? The phrasing is awkward. I think if you used the subject pronouns she and he it would be better, because these describe who got married, but the sentence would still not be clear. How about
He found out she was sick only when they got married, she at almost 18, he at 19.
I think at is necessary to make it grammatical. "He got married at 19" or "She got married at almost 18"* is the "normal" word order here.
*I'm not completely sure about "at almost 18", though - I think this is grammatical, but still a little awkward. It might sound better to just say "17", but it's probably ok as is.