Great results according to your high quality standards, Stu?
Pfft... I have no standards Ayisha.... I just push the little button and magic happens~
I honestly don't know anything about the specifics of the TG-4. I'll send you a PM on FB with the dude's name... You probably already know him since he's something of a fixture around here...
With any small camera, well any camera, if you are going to rely on natural light, then the white balance adjustments are going to fail and fail badly. The "problem" can be reduced with a red filter but really, adding a strobe is the only way to properly get rid of the blue. That's just physics, and not a lot to do with the camera.
And with any camera, get close, then get closer, and stay low. The less blue water that you have between lens and subject, the better. The downside of this technique, which is cited repeatedly by every UW shooter, is that it really limits your creativity. Most new shooters come back with a pile of photos which amount to a catalogue of the critters they saw... an angel fish, a barracuda, a starfish... These images serve the purpose and they are fairly simple to take. But to get to the next level.. the angel fish set against a beautiful chunk of the reef, with your buddy in the background... takes practice, but it also takes hardware... specifically strobes. Smaller point and shoots will always have limitations, but at least the TG-4 has the option of adding a decent strobe, WA adapter etc.
Getting back to your original question about the white balance, when all is said and done, the best that any camera can do is reduce the blueness.... for the simple reason that water is, well, really blue. This can easily be "tweaked" in post-processing with any software.
I think new shooters expect to push the button and get a perfect image and that's rarely the case. All of us that have been lucky enough to get a few "keepers", do "something" with our images before we present them. And this is one of the strengths of the TG-4... it can shoot RAW. While it's easier to just bang off jpegs, the reality is that that jpeg is the result of some software engineer's idea of what that image should look like. Post processing a RAW image will yield a much better final image, but it requires a bit of work.