How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting? The force of gravity is constantly being applied to an orbiting object.  And therefore the object is constantly accelerating.  Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas?  I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it?
 A: The force of gravity has little to do with friction. As dmckee says, what is happening is that the body falls, but precisely because it has enough momentum, it falls around the object towards which it gravitates instead of into it. Of course, this is not always the case, collisions do happen. Also systems of astronomical bodies are complicated and the combined effect of the action of several different bodies on one can destabilize trajectories that in a simple 2-body case would be stable ellipses. The result could be collision or escape of the body. 
In the 2-body case however, the crucial aspect of gravity which guarantees the stability of the system is the fact that gravitation is a centripetal force. It always acts towards the center of the other gravitating mass. One can show that this feature implies the conservation of angular momentum, which means that if the 2-body system had some angular momentum to begin with, it will keep the same angular momentum indefinitely.
(Extra note, even in the 2-body case, there can be collisions and escape to infinity, the first if there is not enough angular momentum (for instance one body having velocity directed towards the other body, like an apple falling from a tree), the other if there is too much angular momentum, resulting in parabolic or hyperbolic trajectories.)
A: Newtonian mechanics explains that they do fall toward the object they're orbiting, they just keep missing.

Quick and dirty derivation for a circular orbit.
Let the primary have mass $M$ and the satellite mass $m$ such that $m \ll M$ (it can also be done for other cases, but this saves on mathiness).
Assume we start with an initial circular orbit on radius $r$, velocity $v = \sqrt{G\frac{M}{r}}$. The acceleration of the satellite due to gravity is $a = G\frac{M}{r^2}$ which means we can also write $v = \sqrt{\frac{a}{r}}$. The period of the orbit is $T = \frac{2\pi r}{v} = 2\pi \sqrt{\frac{r}{a}}$.
Chose a coordinate system in which the initial position is $r\hat{i} + 0\hat{j}$ and the initial velocity points in the $+\hat{j}$ direction. Chose a short time $t \ll T$ and lets see how far from the primary the satellite ends up after that time.
If we have chosen $t$ short enough, we can approximate gravity as having uniform strength through the time period (and we shall show later that that is justified).
The new position is $\left(r - \frac{1}{2}at^2\right)\hat{i} + vt\hat{j}$ which lies at a distance
$$ r_2 = \sqrt{r^2 - r a t^2 + \frac{1}{4}a^2 t^4 + v^2 t^2} $$
pulling our at factor of $r$ we get
$$ r_2 = r \sqrt{1 - \frac{a}{r} t^2 + \frac{1}{4}\frac{a^2}{r^2} t^4 + \frac{v^2}{r^2} t^2} $$
and converting all the $\frac{a}{r}$ and $\frac{v}{r}$ terms into expressions of the period we get
$$ r_2 = r \sqrt{1 - \left(2\pi\frac{t}{T}\right)^2 + \frac{1}{4}\left(2\pi\frac{t}{T}\right)^4 + \left(2\pi\frac{t}{T}\right)^2}$$
Finally, we drop the $(t/T)^4$ term as negligible and note that the $(t/T)^2$ terms cancel so the result is
$$r_2 = r$$
or the radius never changed (which justified the constant magnitude for acceleration, and a small enough $t$ justifies both the constant direction and the dropping of the fourth degree term).
A: I always think of it this way: gravity and the centrifugal force generated by the orbiting object are exactly in balance. If you tie a rope to an object and spin it around you, the centrifugal force will pull on the rope. You pull back with the same force to keep the object orbiting around you. That's exactly what gravity does to orbiting objects.
You can also see that the speed of the object forces it into a particular orbit. If the object slows down, it will fall and either reach a new lower orbit, or crash into the object they're rotating around.
