I would say Newtonian gravity is the limit of general relativity as gravitation becomes weak.
See it this way: the limit when the temperature-amplitude is small of metal extension can be written as:
$$ L = L_0 (1+\alpha \delta t) $$
If $\delta t$ is quite large this equation is no longer applied.
Special Relativity doesn't care about gravity, like the classical mechanics doesn't care about the extension of metal.
EDIT: about the suggestion: where is the Newton's Law of Motion.
First, I think a theory of gravity always has two part:
In Newtonian mechanics, the first part is solved by the Newton's Law of Motion and the second part is solved by the Newton's Law of Gravity. (Although Newton never knew about gravity field, let's just pretend he did)
In special relativity, there may still be the Newton's Law of Motion (modified), and it deals with any kind of motion provided gravity is absent.
In general relativity, there is the Newton's Law of Motion (modified twice), and now it deals with every kind of motion, even in gravity. The most tremendous discovery is the part "How an object creates its gravity field". The Newton's Law of gravity is replaced by the Einstein Field Equations.