For the "public" factor, I think it's rude to publicly chastise an adult as if they are a child. The man is in a war zone and experiencing some real difficulties - frankly, if using a curse word gives him a moments vent, he should curse all he wants (actually - he's an adult, so should curse all he wants).
I can only imagine how hurt he was to read it - because it strikes the reader that the father's primary concern is how the hard time is affecting his son's word choice, not that he's having a hard time, or all the other ways he could be affected by that.
Definitely send a PM to your cousin. Actually, post a follow-up comment - completely ignore what his dad wrote, just say "hey cousin, I can't even imagine how rough it is for you! I'm sorry to hear you're having a hard time. Please know we all love you and are thinking of you!".