I was, and still am convinced I'm NT, even if that's not the case I have no reason to be diagnosed, but I'm still diagnosed so I don't think you should risk your navy career on just trying to prove your NT.
Also I wouldn't be too sure about private medical records finding their way into the hands of the navy, there probably is a system in place for them to get their hands on them and lying on the navy form is a big risk, it will ask you directly if you have autism or other social disability (been there, done that, got the boot).