Yes, every gravitational force in Newtonian mechanics has an equal and opposing force, and it usually acts on other mass.
More specifically, every two pairs of masses feel a gravitational force that's proportional to the product of their masses and inversely proportional to the square of their relative distance, but more important is the fact that both masses feel the attraction to each other.
Thus, when you throw a ball of ~100g in the air, it experiences a gravitational force of 1N downwards, and in doing so it exerts a force of 1N upwards on the Earth. The reason you don't observe the Earth moving is that its acceleration is so small (on the order of 10-25 m s-2) that it gets swamped in everything else, but it does happen.
Now, it's important to note that gravity is not usually the only force acting on any object at a given time. If it is, then the total force will be nonzero and the object will accelerate (as per Newton's Second Law). Conversely, if an object is not accelerating, then the net force on it is zero, and there must be additional forces that cancel out the gravitational one.
For a book lying on a table, for example, the weight is cancelled by the upwards reaction force from the table. (And, of course, this gives an added reaction force downwards from the book on the table, which gets cancelled by a correspondingly larger reaction force from the floor on the table.)
Similarly, the reason that masses (like, say, the interior of the Earth) don't get compressed any further is that any given volume of rock will be acted on by the downwards gravitational force and by the upwards pressure from the rocks below it.