Is Spivak right in what he says about Galileo?

On chapter 9 of M. Spivak's book on calculus there is an exercise in which Spivak asks the reader to prove that Galileo "got his facts wrong". More specifically, Spivak asks one to to show if a body falls a distance $d(t)$ in $t$ second and $d^{\prime}$ is proportional to $d$ then $d$ cannot be a function of the form $d(t) = ct^{2}$.

Settling it is kind of a no-brainer: yet, did Galileo really claim what Spivak is attributing to him therein? Do you know if this "mistake" by Galileo had been noticed before? If I understand correctly, even Newton took for granted the claim by Galileo according to which "the descent of bodies varied as the square of the time" (cf. p. 21 of vol. I of the University of California Press edition of the Principia)? What's going on here?