I live on Oahu, husband is in the Navy too. I'm going to Tripler for a natural childbirth (hopefully) , and they've been surprisingly open minded about everything. I think it depends on how natural you're trying to go... there are certain things they kinda insist on, like they'll do intermittent fetal monitoring, etc, but if everything is going well, they will pretty much leave you alone (from what I've heard from another natural childbirth mom).
The one big thing at Tripler that's great is that they are very well versed in the fact that you have the right turn down any treatment. It's really been very empowering for me to hear over and over again that, "Well, this is the way we like to do things, but if you feel differently, that's your right. We can't do anything to you or your baby without your okay." It's a bit frustrating at times to have to push back against their recommondations but ultimately, the choice is yours.
I live on the Windward side and my Bradley instructor says that Castle Hospital here is also welcoming for natural births -- she had all four of her kids naturally there. They have more of a "birth center" set up attached to the hospital that you might like. It's our backup hospital in case we can't make it out to Tripler for some reason.
I know it's too late for the original poster but hopefully it helps someone else!