general relativity - Doubt on Newtonian weak field metric, accelerated frames and metric tensor transformation

Suppose we do not have yet General Relativity conclusions (like, Schwarzschild Gemetry and Weak Field Approximation) , but rather, just Minkowski space-time, newtonian gravity, principle of equivalence and special relativity on accelerated frames (i.e. special relativity on non-inertial frames).

First, we have then the Minkowski spacetime without any gravitational influence:

$$ds^{2} = -c^{2}dt^{2} + dx^{2}+dy^{2} + dz^{2} \equiv \eta_{\mu\nu}^{(Far-from-Gravitational-field)}dx^{\mu}dx^{\nu} \tag{1}$$

Secondly we then have a spacetime, which descrives the effects of Newtonian Gravity:

$$ds^{2} = -\Big(1+\frac{2\Phi(x',y',z')}{c^{2}}\Big)c^{2}dt^{2}+\Big(1-\frac{2\Phi(x',y',z')}{c^{2}}\Big)(dx'^{2}+dy'^{2} + dz'^{2})\equiv g_{\mu\nu}^{(Under-the-Gravitational-Field-near-Earth's- Surface)}dx'^{\mu}dx'^{\nu} \tag{2}$$

Now, is it possible to say that the spacetime which describes Newtoninan Gravity is obtained by just a coordinate transformation between an inertial frame to an non-inertial frame (Much like from Minkowski spacetime to Rindler Spacetime)? I.e. is the Newtonian Gravity just another effect of a "accelerated reference frame" (then here we see the principle of equivalence)? :

$$g_{\mu\nu}^{(Under-the-Gravitational-Field-near-Earth's- Surface)} = \frac{\partial x^{\alpha}}{\partial x'^{\mu}}\frac{\partial x^{\beta}}{\partial x'^{\nu}}\eta_{\alpha\beta}^{(Far-from-Gravitational-field)}$$