If the gravitational acceleration toward the moon was the same regardless of where you were, then the whole earth+ocean system would accelerate together, without leaving any internal evidence of the gravitational effect. That's sometimes called the principle of equivalence. However, the gravitational pull toward an object is bigger the closer you are to the object. So the part of the ocean closest to the moon is pulled a little extra, compared to the average of the whole earth. It bulges up. The part farthest from the moon is pulled a little less than average, so it bulges the opposite way- which is also up, on that side of the earth.
The same thing happens from the pull of the sun, although not quite as much. During full and new moons, the moon and sun effects add together to make extra-big tides. During half-moons, they tend to cancel, making smaller tides.
What could have possibly led you to claim "I have never believed the moon's gravity pulls the water..."? If the sun didn't pull on the water, wouldn't it keep going in a straight line and leave the sun-orbiting earth behind? If the earth didn't pull on the water, wouldn't any little splash launch water into space? Why would there be any water left here? And if the sun and earth pull on the water, why wouldn't the moon also do so? Do you seriously think that Newton and every other physicist and geologist since then has been completely clueless?
(published on 03/10/2011)