I understand the privacy issues, but I think that ceases when the person dies. In this case, the Army will give his parents all of his snail mail letters that he had in his possession, so why is email any different? Also, if they are his heirs and executors (which it sounds like they are), they are entitled by law to his medical records. What's more private than that?
Email is different because it carries the expectation of privacy. You say things or communicate with people with the expectation that your words are yours and no one elses. If you save letters from someone and put them in a box, you expect that others might eventually get around to seeing those. The same is true if you print out email messages and leave them with your possessions.
Privacy never ceases when a person dies - they take their memories and thoughts with them. Email, with its expectation of privacy, is often an extension of those.
The fundamental question is this - do we know for certain if he wanted to divulge everything in his account to his family?
If we don't, we should err on the side of his privacy. It's for the sake of his dignity.