This might not be the right place to get an objective answer to this question, as I assume most of us value sound higher than the average person, but could we even say sound is just as important as what we see, and if we take away or change the sound from a movie, the overall experience will be changed completely?
I am of the belief that in contemporary media sound is just as important, if not more important than it used to be, as well as equally important as the visual.
You can have the most impressive, most advanced graphics in the world (take Avatar, or Transformers), but without sound that is equally as impressive a feat, you can't sell the graphics, and the viewer is taken out of the experience.
The more that big pictures up the ante with visual effects, the more sound has to step up to match it. Think back to when Star Wars came out. It was incredible. Nobody had ever seen anything like it before. Nobody had really been a sound designer before either. If they had just put every day sounds to Star Wars, the audience wouldn't have believed it. What Ben Burtt did was create sounds than nobody had ever heard before to go with the picture that nobody had ever heard before. That was the only way to make it remotely believable. And what a fantastic job he did!
Now that anywhere from 30% to 100% of the picture is CG in most blockbusters, sound is the "Glue" that holds the picture together. A giant CG robot can only look so lifelike. It can't be perfect. What takes your mind off of the imperfections is the organic sound it makes.
Stepping away from movies from a minute and touching on mobile and internet media, I think sound is just as important. Sound has become a sort of tactile feedback. You roll over something on the internet or press a button on a device and it makes a sound, however subtle it might be. You take that away, and it completely changes the experience. Sound is immersive, it's what completes an otherwise incomplete media experience.
My 2 cents
Colin makes some great points, the most important of which that context is supremely important. I've seen films so visually arresting that the sound wasn't needed for long stretches, and I've seen films (animated, usually) that would have been meaningless without sound.
Sound adds context to the image, and vice-versa. I think it's hard to not just watch visuals without sound, but also to listen to sound without visuals (which I often do, when I study film sound design). The mind craves both to make sense of a moment, in media and real life - we're evolutionarily hardwired to do so. Both will change the emotional content of, and the subjective "truth" of, a moment. The best image/sound artists know how to exploit this without seeming overly "clever" (I'm looking at you, Atonement).
Outside of film and games, sound is almost always the bastard child of temporal media. No one budgets for sound in an interactive piece and then complain when it's absent. Of course, the same is true of film/game directors/producers who don't budget well and then complain about sound quality.
Our ears and eyes both operate logarithmically, but our ears have far greater dynamic range than our eyes. There are some psychological studies that suggest that auditory input can significantly enhance long-term memory, thereby helping with the sublimation of conscious learning into intuition. I'd love to understand that more, as I could use more space in my brain that isn't taken up with advertising jingles and Monty Python sketches.