the gravity has nothing to do with the earth spinning. you have to think of gravity as an atraction between you and any other object.
between any two objects there exists attraction in the form of gravity. even between objects of relatively small mass/size, there is little gravitational attraction, but it is most deffinately there. right now, you and your computer are pulling on one another through gravity.
now think of that in terms of a whole planet...from your perspective, it seems like you're on top of the world, or the world is completely beneath you. That means you have the entire mass of the planet earth pulling you towards it, and that's why it seems like gravity means that objects fall down (because from our perspectice, they do! but from space, its just the object being pulled to earth)