As Marshall McLuhan said, "The medium is the message."
While I'm not so sure he was talking about underwater photography, the principle is similar. The medium in which a signal travels influences the information received. This not only happens underwater, but topside is well. The sky tends to be blue (where you dive...not here in Rochester, NY, where gray is the dominant color) because of the differing interaction between atmospheric molecules and wavelengths of light.
Underwater, a similar effect occurs, and is more dramatic. Sunlight entering the water from above becomes filtered by the water, and the reduction of red light is greater than that of blue and green. So the deeper you go, the percentage of red in the available light becomes less and less. Additionally, the distance between you and what you are looking at provides an additional filtration, removing even more of the red light.
Actually, your camera (without flash) does a pretty good job of capturing the scene. The blue/green cast you see in your pictures is a good representation of the available light that bounced off the subject and was captured by the sensor. But your brain does a really nice job of automatically balancing those colors into a much more pleasing color balance. "Eyes, this can't be right. I'll fix it for you. You're welcome, Brain." It auto white balances for you, without any conscious involvement. So what your camera records is not what you saw.