This is just totally wrong. If there were really multiple forces acting on the moon which all canceled out exactly, the moon would move in a straight line. This is the essence of Newton's first law (although we can see it just as well by looking at the second law).
Ignoring the (small) effects of the rest of the solar system, the only force acting on the moon is the gravitational force between it and the Earth, which is directed along a line between the centers of the two bodies.
The confusion here stems, not from an additional force, but from a misunderstanding of the connection between forces and motion. Newton's second law tells us that the sum of all the forces acting on an object will be proportional to its acceleration. In other words, forces change motion. In this case, since gravity is attractive, the basic change in the moon's motion will be for its path to curve towards the Earth instead of remaining a straight line (which it would be if there were no forces).
In Newton's theory of gravity, it turns out that there are four different shapes that an objects orbit can take, depending on how fast it's moving and how close it comes to the gravitating object. These, however, are relatively difficult parameters to use, so we generally talk about the energy and angular momentum, instead (but, we could transform directly from one of these sets of parameters to the other).
For any given angular momentum, the lowest energy orbit will be a circle. All orbits between this energy and a mechanical energy of 0 will be elliptical. 0 energy orbits are parabolas and posive energy orbits are hyperbolas.
From this, it's clear that any orbit with negative mechanical energy (or, equivalently any bound state orbit) will be a closed path. So, no orbits will lead the moon to progressively spiral towards the earth. If its orbit is already large enough that it doesn't hit the earth, it will stay that way.
To understand why these stable orbits are allowed, we can think about what physically happens in each type of orbit. First, we consider a circular orbit. In this case the object is always moving perpendicularly to the force of gravity. This means that the object must have just exactly the right velocity that it will always fall towards the ground at just the same rate that the ground falls away below it, due to the curvature of Earth's surface.
An elliptical orbit is what happens when the velocity is not just right for that to happen. Let's say it starts off moving too slowly. Then, as it falls in its orbit it gets closer to the earth. But, as it gets closer, it must also speed up due to the conservation of energy. The closer you are to a gravitating object, the more negative your gravitational potential energy becomes. So, for your total energy to be conserved, kinetic energy must increase, meaning increased speed. Eventually a speed will be reached such that the object is falling slower than the ground curves away below it. At this point, it will start moving farther away from the surface. At least until it reaches a point when it is too slow.
The essence of this argument comes down to the conservation of the orbiting body's energy and angular momentum. Only if there is some outside interacting which progressively changes one or both of these parameters is it possible for a stably orbiting body either to crash or to escape.
