Actually, the Navy (I was there.)
Radiation from natural sources is ignored; radiation from Navy reactors and related sources is all important.
Example: a sailor took his TLD home on leave (personal dosimeter, attached to your belt. You don't think about it.) His parents ran a veterinary clinic that had an old fluoroscope. When that TLD was read at the end of the month all hell broke loose, resulting in a new Navywide rule that TLD's be turned in to (and signed for by) the officer signing one out on leave. The main concern was proving that the exposure wasn't from a Navy source; hanging around that clinic might have been... unwise, but wasn't a problem for the Navy.
I was underwater for Chernobyl, and not scheduled to get surface air for a month or so. We were concerned that when we did, we might 'suck in' some radioactivity and set off alarms (yes, they were that sensitive.) If that happened, everyone in the ship would have to wear respirators for an hour or two while we proved it wasn't our reactor, after which we could relax and breathe freely, radioactivity and all. (Nothing happened, but we were standing by.)
Why? Not so much legal liability (though I'm sure that's considered), but the Navy's delicate relationship with the NRC. As one senior officer observed, during any given Christmas week there were at least a dozen reactors floating in the river at Norfolk, tended by a couple of (admittedly highly trained) 20-something high school graduates and one sleepy officer and CPO. No one gave that much thought, but imagine the outcry if someone suggested building a commercial reactor nearby (with much greater oversight and safety features than a submarine) to provide power to the city. The Navy, by virtue of its overachieving training, documentation, and safety programs, not to mention Cold War precedents and institutional secrecy, gets to run its reactors without NRC or much civilian involvement; anything that goes wrong and reaches the press threatens that arrangement, without which the program realistically couldn't exist.
I'm not complaining or trying to blow some kind of whistle, BTW: the program works. I probably averaged less rads underway than on a sailboat; certainly less than on a fossil fuel fired ship. I don't live in Denver but I would, and I don't worry about chest x-rays or long airline flights. I'm glad the Navy took good care of me, but I also understand their reasons.