There is no derivation of Hooke's Law from Newtonian mechanics. By Hooke or anyone else. At the time of Hooke, this was an experimental law. If Hooke had some theoretical explanation, it was incorrect. The derivation of this law from other more fundamental laws is a matter of Solid State physics, and I do not think this is achieved even today.
Remark: If "Newtonian mechanics" means the contents of his book Principia, then Hooke's law was published earlier.