No, they were not 'both right and both wrong.'
One of the less-appreciated themes of this film was the issue of "doing the right thing." Now, we all know "doing the right thing" is an ambiguous concept - the "right thing" is more a concept, one that is subject to the totality of the circumstances surrounding a decision that must be made.
I am unfortunately not well-versed in nuclear weapon employment procedures for U.S. fleet ballistic missile submarines, but even if the procedures that were depicted in the film (particularly the one regarding the lack of a contrevening order) were true-to-life, Captain Ramsey's actions were not only wrong, but imprudent and he would not be able to use the excuse of "I was just following orders" as a means of covering himself.
In the military, or anywhere else in life, for that matter, using your better judgment and looking at the big picture is an imperative for survival. The thing that, I believe, makes humans unique, is our ability to overcome our natural or artificial instincts, habits, etc. and do what we know needs to be done, no matter what the odds or obstacles standing in our way. On a procedural basis, Captain Ramsey may have been "right" and that was probably with the tribunal at the end was referring to in the now-famous declaration. However, procedural correctness does not always equal ethical, moral, managerial or operational correctness. The release of nuclear weapons is not your typical military decision. It is a decision that carries a weight that the world cannot carry and comes with consequences that can not be dealt with. As Commanding Officer of the most lethal weapon in recorded human history, nobody should have known that better than Captain Ramsey.
This thread is not about Captain Ramsey's motivation(s) for behaving in the manner that he did. This thread assumes that Captain Ramsey's motivations were pure and that his heart was in the right place. Even when we give him that benefit of the doubt, Ramsey's actions were inexcusible. His behavior showed little to no acknowledgement of what he and his crew were about to undertake and his refusal to have the communications problem looked into was incredibly imprudent and showed extremely poor judgment. Captain Ramsey's actions and behavior showed very little regard for the consequences and implications regarding the use of nuclear weapons, which is conduct unbecoming of someone who is tasked with such a responsibility. Ultimately, Captain Ramsey, along with his lackeys, who were just as culpable, all failed to look at the big picture, grasp the "point" of who they were and what their duty truly was and they made the wrong decisions.
I am not one to challenge Navy flag officers, but the tribunal is very wrong. Captain Ramsey was not right and he failed miserably in his duty as Commanding Officer of a Fleet Ballistic Missile submarine. The tribunal also failed - they attributed procedural correctness to Captain Ramsey being simultaneously right along with Lt. Commander Hunter. This implies that the top Navy brass does not hold Commanding Officers to a high standard, if all they expect is procedural correctness. That may fly with personnel lower on the chain-of-command, but it does not fly with servicemen who are responsible for hundreds of people, millions of dollars worth of equipment and who have the ability to make large-scale decisions of an existential nature.
This is what I had on my mind today. Great movie, a lot to think about.