For a mass m to orbit earth with mass M at radius R with velocity V, the centrepital force has to equal the attractive force or

(mV^2)/R = GmM/R^2 for the situation where the attractive force decreases as 1/R^2. G is the gravitational constant.

Solve for the Velocity versus radius to get

V^2 = GMR(1/R^2) as the condition for orbit

You can replace the (1/R^2) with (1/R^3) or (1/R^4) or any other well behaved function f(R) that goes monotonocally to zero faster than 1/R and the equation can be solved (at least numerically) for the Velocity needed for any radius R

So you misunderstand what is meant by stability.

You can have an orbit existing with any circularly symmetric force. It can be a nice simple function of r or it can be a horribly complex one complete with discontinuities (as long as you aren't doing it at the discontinuity). If you stick in a value of r you will get some force at that point, you then find the required velocity for that force and radius and you are done.

But this gives a perfectly circular orbit. The issue is what if this orbit is perturbed or you don't start off with a perfect circle, i.e. what if you have a slightly higher or lower tangential velocity, or if you have any radial velocity?

For an unstable orbit, a slight change can result in the orbit collapsing.

A real life example of this is the L1 point, which also links to magnetic levitation.

The L1 point for an orbit is where you have a small body perfectly balanced between 2 larger bodies, e.g. a small satellite between Earth and the sun.

The forces work out to give a system where the object remains in place. For the L1 point this means that as the object orbits the sun, the Earth provides enough force outwards to slow the orbit such that it keeps the same position relative to Earth such that the sun, object and Earth are all aligned.

For magnetic levitation you have it so the magnetic force just cancels out gravity pulling it down so it stays put.

The problem with both is that they are inherently unstable.

If your satellite moves closer to the sun, then it will need a corresponding stronger force from Earth, but as it moves further away from Earth it gets a weaker force from Earth and thus the satellite moves from its position and starts just orbiting the sun disconnected from Earth.

For the magnet example, if it gets closer to the magnet the force increases and it flies up to the magnet; if it gets further away the force decreases and gravity wins and the object falls.

Both of these are intrinsically unstable. While you do have a perfectly balanced point, any slight perturbation will result in it going away from that point.

For a stable orbit the opposite would happen, you have a perfectly balanced point and any slight perturbation (note: slight, not massive), will result in it going back to that point.

For example, with a gravitationally bound orbit around a large body, you have the hypothetical perfectly balanced circular orbit. If you are going too fast then you go to a higher orbit, but also slow down as gravity would then be pulling you back, not just inwards. This results in you reaching a peak where you are going too slow for the orbit and thus you start falling down. Gravity then speeds you up and you end up going back up, oscillating between 2 radii in a stable orbit.

As a simple analogy, a stable orbit corresponds to suspending a long rod from the top, an unstable one would be having it stand vertically on a tiny ball.