Alan,
From
"The Cambridge Companion to Newton" provided here by Stephen, it is visible Newton "experimented" (mathematically) with movement of bodies influenced by central force between them that has various dependency on distance r between them. We would say today he played with different math models of orbital motion, by varying the value of exponent to "r" (distance between bodies) in his equations.
Basically, he concluded then that gravitational force obeying only inverse square law (1/r^2) results in stable elliptical orbit (circular is a special case of ellipse where eccentricity is zero), which corresponded with observations.
If gravitational force obeyed for example 1/r^3 law, the orbit would have been inward spiral (I don't remember which type of spiral exactly, it could be Archimedes' ?)... resulting in collision of two bodies. BTW, spiral (and unstable) orbit results from any value of exponent >2.
If exponent applied to r is less than 2, then spiral would have been "open", resulting in increasing separation of bodies (and escape).
Remember, this was happening in times physicists did know how gravity force behaves with distance... hence the discussion.. involving mathematical details... So, apart from historical insight, it is irrelevant to our discussion here.
However, my question to you was:
"why and how spiral motion causes
precession".