If this marine had wanted his family to know the contents of his email, he would have CC:'d them. Why stop at electronic mail? Should the family find out about snail mail correspondence their son was carrying on, would they demand that the recipient forward copies to them?
Private correspondence is just that. Had their son wanted them to be in the loop, they would have been.
Let's get his phone records; let's see if we can get the text messages from his cell phone......let's do everything we can to learn about our son now that he is dead, since we never bothered to know him when he was alive.
End of Story!