Anyone that has access to starlink can track their vessel.
Regarding tracking – and I don’t know if that’s the main concern here – it won’t just be the Starlink transport itself. My guess is that they’re more worried about having devices connected to the Internet.
Let’s say that I have a smartphone. I download an app because it’s got a fun game attached. It comes up with some dialog about permissions when I installed it, which I promptly accept and forget about. This includes location permission. A month later, I get deployed, and I put my phone on the Wifi network that the nice ship IT person has set up that we all chipped in for. Yay!
So now, every now and then, maybe that app is phoning home and reporting my location using the GPS sensors in the phone. Now, maybe that game company is on the up-and-up. Maybe they’re selling data to some data broker. Maybe they’re in a country subject to legal requirements to turn over data to the government on demand. Maybe they’re securing their own systems to a level sufficient to keep out nation-state level intelligence agencies, and maybe they aren’t.
But my guess is that the Navy doesn’t want to deal with those possibilities.
And that’s even before potential for things like microphone permissions for a device that’s on a warship.