Derive metric from weak gravity case?

Gravity seems to be stronger in larger distances and weaker in smaller distances.I am refering to the model of gravity where the gravitational potential is independent of distance , the force scales inversely with distance and Newtonian gravity is a high energy solution.

The metric for Newtonian gravity can be approximated as a linear sum of the Minkowski metric and a metric which includes the Newtonian gravitational field.However for the case of weaker field density of gravity im talking about what would the metric look like?Linear again right?Because you cant go below linear or something else entirely?Thanks.