In another question How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting?, one can read an affirmative answer. They how do you explain satellites falling to Earth? (Here, falling means colliding with the Earth surface or burning out before reaching the surface. For example, fragments of Sky Lab fell on Australia in 1979.)
In the question you posted the link of, the mathematics is given by the chosen answer.
In the second answer there, the importance of conservation of angular momentum is stressed. In general, when considering gravitational solutions one has to think of conservation laws.
As an answer here stressed, friction in the remnants of atmosphere reduces the kinetic energy and also angular momentum of the satellites and they may eventually fall to the earth. ( conservation of energy).
It is instructive to examine another instability, when angular momentum is changed, which happens with the tides in the earth/moon system. The effect of the gravitational interaction is to increase the energy of the moon, which goes into a higher orbit; i.e. the complex gravitational interaction transfers kinetic energy to the moon. If you read the article you will see that this angular momentum change due to tides induced on the earth can also dominate the path of artificial satellites.
The moral of the story is that interactions determine the stability of orbits in gravitational systems, which change following conservation laws.
First of all, even if satellites that orbit the Earth do follow, on a good approach, the newtonian physics, they are also subject of several other interactions like moon's influence, non-eshpericity of the Earth, gravity anomalies, spacial dust, etc. All those interactions cannot be predicted for long enough times, most of the time because those interactions can lead to chaotic behavior.
That said, most of the satellites have some mechanism to re-adjust its orbit, like some active feedback trusters. As you might have already understood by now, sometimes these mechanisms fail, causing the satellite to crash into the Earth.
Usually this is done on purpose when the satellite is no longer useful, in order to prevent it from becoming a hazard to other satellites and/or spacecraft. This was, for example, the case with Skylab.
Satellites in lower orbits can also be slown down by atmospheric drag. If the satellite is for any reason no longer able to compensate for this, it will slowly drop into a lower and lower orbit, eventually reaching the point where the atmosphere is thick enough to burn it up or (on rare occasions) hit the Earth.