Since Sir Isaac Newton invented gravity, "Newtonian gravity" is a well-defined notion, i.e., gravity as per Newton. Thus, we are free to talk and to mean whatever we want discussing Galilean, Aristotelian, Keplerian, Pythagorean, Copernician, etc. gravities.
What is attributed to Galileo is a notion that a constant velocity does not modify bodies' behavior. This means that in a system S' that moves with velocity v relative to a system S along x-axis,
x'=x-vt
t'=t
Any model of anything which assumes this coordinate transformation, is Galilean. Newtonian is such. GR is not.