What should he have done? Clearly it was his duty to inform the outbreak of a contagious and debilitating disease on his ship be made known to senior officers. Indeed given that the CVN is a capital ship and major national strategic asset he was expected to do so in a matter that would come to the attention of the Chief of Naval Operations and the Secretary of Defense, and even of the President (Who is referred to as the National Command Authority). He was in fact expected to seek relief for the ill and their isolation or removal to protect the other crewmembers. Above all his duty was to keep his command combat ready or support measures to restore it when it was unable to fulfill any assigned mission. This is the key. His first duty was to his mission. He was expected to care for his troops and inspire their loyalty but not at the expense of the mission.
How should he have done that? By sending a classified message through his chain of command that informs all concerned parties and is marked of such urgency that it will come to the attention of the NCA. The means of doing so is in a formatted message called a Casualty Report or CASREP. It is the same method used for reporting a broken part. The urgency of the problem can be marked in the message in a manner designed to get it responded to effectively. There are additional message formats the Captain could have used within the system. He also could have sent a personal message to the CNO or Secretary of the Navy if he thought that his needs were not being addressed and a serious threat to national security was being created by the neglect of his seniors. All of these messages would be classified.
What he did instead was to send an unclassified letter stating that the health of the crew took priority over any operational mission. That letter which was certain to become public informed all potential or actual enemies of the United States that the Theodore Roosevelt was not combat ready.
There is a reason that officers are told not to get to close to their troops. An Infantry officer once said to me, "Don't use their first names. You may have to send them to their deaths." The Naval Officers Guide warned that even if you had the gift connecting informally you had to be careful because you must not make it harder for the next guy.
I am sure that I would have felt the urge to support him. It is difficult. We need not accuse him of wanting to be an American Caesar without more proof, and the military is cursed with to many gray technocrats who cannot Lead or do not care for their troops but his use of the line "we are not at war" in his letter was I fear inexcusable.
It was called "Overidentification with the men."