How to fix white balance gone bad?
This is not the auto white balance screwing up.  This is the way that SCUBA footage looks without artificial light.  Even shooting with my SeaLife DC1400 which has specialized underwater white balance settings, you get a strong color cast, particularly in green water.  

Water doesn't block frequencies of light evenly and as depth increases, much of the red light is lost from the spectrum.  Unless you have an artificial light to put red back in to the color, there is no signal there to be had.  The color cast comes from the color of the light exceeding the sensitivity of the sensor to be able to adjust for the imbalance between the red, green and blue light.

Update: Ok, I just looked at your footage and it looks like there may have been a technical glitch with it.  That is extremely green, even for underwater footage.  That almost looks more like a malfunction, but there still isn't enough information there to correct.  It looks like red and even blue data is almost completely absent from the recording.  I guess maybe my SeaLife DC1400 does a better job than I thought with color if there wasn't a technical problem there.

As far as ways to try and salvage it, you could go black and white with it or could try reducing the amount of green in it using a hue adjustment, but it still is only going to look "less bad" not "good".  Wish I had better news for you, but it kind of goes with the territory of shooting underwater at any kind of depth at all without a light, particularly in colder green waters.