I've dug up the following from
The Cambridge Companion to Newton:
The spiral orbit with a center of force at its pole is a particularly simple direct problem whose solution is an inverse-cube force [see attachment 1, the first paragraph of which is from an earlier footnote, but still interesting]. In this way Newton could have discovered in a straightforward manner that for the inverse-cube force the orbit reaches the origin “by an infinite number of spiral revolutions,” as he described it in his 13 December 1679 letter to Hooke. It is noteworthy that in “De motu” Newton quoted this result in a scholium without giving a geometrical demonstration as he did with his other propositions, and later on in the Principia, he applied the 1/r3 force law rather than the physically more interesting 1/r2 case, to solve explicitly the inverse problem (see Theorem 41, Corollary 3, Book 1) [see attachment 2]. Although Newton could also have applied his curvature method to solve the case of an elliptic orbit, there is no direct evidence that he actually carried out such a calculation.
The missing ingredient for a complete solution of the orbital problem, which must include the temporal as well as the spatial dependence of the motion, was provided by the fundamental idea of Hooke to view orbital motion as compounded by a tangential inertial velocity and a change of velocity impressed by the central force. This idea can be expressed in simple mathematical form for forces that act
as periodic impulses for which the curvature method is not applicable, and it leads directly to the area law (see Principia, Proposition 1, Book 1). After the correspondence with Hooke, Newton evidently understood the equivalence of these two distinct physical approaches to orbital motion, but he never credited Hooke for his seminal contribution.
Brackenridge, J., & Nauenberg, M. (2002). Curvature in Newton's dynamics. In I. Cohen & G. Smith (Eds.), The Cambridge Companion to Newton(Cambridge Companions to Philosophy, pp. 85-137). Cambridge: Cambridge University Press. doi:10.1017/CCOL0521651778.004