Because the vertebrate eye evolved in the sea, it makes sense that it was selected to be sensitive to those wavelengths that pass through seawater. This idea is supported by noting that many deep-sea organisms are less sensitive to red wavelengths (which get absorbed preferentially). Since the wavelengths we see are those that pass through water (and air), we perceive both media as being transparent. 

What Mike said (as usual)!

Also a few thoughts, possibly somewhat off topic:

The nature of water and of what we call visible light, and later of air, had a great deal to do with it too. Consider:Water and air actually absorb wide ranges of wavelengths, but both of them have a broad "window" of transparency around the range of wavelengths to which our eyes are sensitive. Our own vision cuts off a little short of the ultraviolet, which some birds and insects for example can see, and some eyes can see further into the infrared then we can. But generally our window of visible wavelengths matches that of the transparency of water and air pretty well. (Bear in mind that the air that we breathe is generally pretty moist as well.)

This suggests that once we start genetic engineering to produce Homo superior, one of the first things we shall do well be to add another octave of visible wavelengths at either end of the limits to our vision. While we are at it we had better add bigger, and possibly more, eyes, together with the necessary mental equipment for exploiting them.

Now, notice a few conservative notes to what I said:For one thing, while I was at it, why not go all the way to x-rays at one extreme and short waves at the other? There would be fascinating and useful things to see and all points of that spectrum. I am reliably informed that Superman, for one, can in fact see in all those wavelengths and in fact x-rays as well.

It is not fully impossible, but the organs we would need would not be our eyes. For most of the range they would not resemble eyes at all. Consider infrared. The most sophisticated infrared eyes that I can think of offhand are the pits of pit vipers such as rattlesnakes. They behave more like pinhole cameras than eyes. Also, they work best at close range when the snake is cold and the prey is warm. They do not form a clear image. It is in fact possible in principle to form better images in infrared than the best that snakes can do, but it probably would take tens of millions of years of selection to achieve major evolutionary breakthroughs to do so through natural selection. For mammals to do so would be more demanding. We produce a great deal of infrared fog ourselves. One reason we can see so conveniently in what we call visible light is the fact that our bodies do not produce visual noise in those wavelengths.

Another reason is that visible light behaves well with lenses of materials and sizes that our bodies can produce precisely and efficiently. This is less true for most of the wavelengths that we miss.

Not to make too much of a meal of it, nor to generalise too strongly, the mechanisms of our eyes reflect the medium in which we live, the available light in our environment, and the range of contiguous wavelengths that efficiently form images in the materials that our bodies can produce and manage.

There also is the question of "evolutionary opportunism", the options open to natural selection for producing different kinds of eyes. This is a very difficult thing to do. It looks suspiciously as though all the rhodopsin-based (basically, pigment-based) eyes in nature stem from a single ancestral development, whether the eyes of worms, clams, spiders or humans. That is a very sobering thing to consider. If we want to make major strides in improving our vision, then natural selection will not be the appropriate route.

There is a big factor not addressed above. Vision depends on external natural illumination (at least while the eye was evolving).

We happen to live near a star of spectral type G2V that behaves as a black body at 5800 deg C. It radiates strongly in specific wavelengths (100 nm to 1500 nm) with a strong peak at 500 nm, and a wide (but low-power) infra-red band up to 1000000 nm.

It happens the atmosphere absorbs almost all the UV below 320nm, and IR is very low power at each wavelength, and also has many absorbtion bands. The wavelengths that reach the surface are almost entirely in the "visible" band (unsurprisingly).

Vision in water relies on the coincidence of the Sun's spectral range with non-absorption bands in both air and water. Without this double window, it would be dark underwater at all wavelengths, and the eye would not have evolved.