Are there good reasons why special relativity should motivate geometrised gravity in a way that Newtonian mechanics does not?

I have studied a bit of Newton Cartan theory recently, the geometrised version of Newtonian gravity in which gravity is due to the curvature of spacetime, but is Newtonian (simultaneity is absolute).

This theory allows us to fill in a table of theories in which we have geometrised vs non-geometrised gravity as the columns, and the rows are absolute time vs constant speed of light. Such a table would look like

$$\begin{array}{cc} \textrm{Newton-Cartan} & \textrm{Newton}\\ \textrm{GR} & \textrm{SR} \end{array}$$

On consideration of such a table, I wondered why I always thought that the motivation for geometrised gravity came from relativity. Clearly they are independent constraints on a theory, since every box of the table is filled.

So, my question is whether there are good reasons that special relativity should motivate geometrised gravity in a way that Newtonian mechanics does not.

• I'm not so sure that your 2x2 table makes sense. Newtonian gravity and Newton-Cartan are two different formalizations of the same theory. They make the same predictions. GR and SR are not the same theory. SR is a special case of GR. – Ben Crowell Jul 23 at 12:26
• It is debated whether Newtonian gravity is equivalent to Newton-Cartan, i.e. do they agree as to what frames are intertial? Nonetheless, SR is a theory without gravity unlike the other three. I guess you could interpret the columns as curved vs flat connection? – Joshua Tilley Jul 24 at 13:58