Is there an accepted axiomatic approach to general relativity? I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since other books I have read haven't emphasized it as much. 
My Question:
Is there an accepted set of axioms or principles that constitute the core premises of GR from which many, most or all relevant properties can be deduced? 
 A: General relativity can be constructed from the following principles:


*

*The Principle of Equivalence

*Vanishing torsion assumption ($\nabla_XY-\nabla_YX=[X,Y]$)

*The Poisson equation (or any other equivalent Newtonian mechanics equation)
Explanations:


*

*The Equivalence Principle can be used to show that spacetime is locally Minkowskian, i.e. the laws of special relativity hold in an infinitesimal region around a freely-falling observer. This is equivalent to the mathematical idea that a manifold of dimension $n$ is locally homeomorphic to $\mathbb{R}^n$. This allows to do two things (that I can think of at the moment). We conclude that spacetime is a manifold. We also can make the substitutions $\eta\rightarrow g$ and $\partial\rightarrow\nabla$, which yields the correct (there are exceptions) GR equations.

*This is required for the geodesic equation to be obtainable from a variational principle because it implies the Christoffel symbols are symmetric. This condition is relaxed in certain theories such as Einstein-Cartan theory or string theory.

*Simply put, we need this equation to fix the constants in Einstein's equation. 
All treatments of GR use the Principle of Equivalence. Weinberg's treatment especially so. The reason for this has to do with his background as a physicist. Weinberg was (and is) one of the greatest particle physicists alive. His dream was to write a coherent quantum field theory for gravity. In his mind, $g_{\mu\nu}$ being called the metric tensor is an "antiquated" term left over from when Einstein learned differential geometry from his friend Grossmann and Riemann & co.'s old papers$^1$. In Weinberg's mind, $g_{\mu\nu}$ is just the graviton field, and any connection to geometry is purely formal$^2$. In texts such as Carroll, Straumann or Wald, they use the EP to make the connection
$$\tag{1}\text{Equivalence Principle}\implies\text{Spacetime is a manifold}$$
From that point on, spacetime being a manifold is assumed. Weinberg, however, was of the opinion that gravity had nothing to do with geometry and manifolds and this mathematical description was a pure formality. He has to stress the EP because philosophically he didn't accept (1). 

$^1$ See the first paragraph of section 6.9.
$^2$ See, for instance, page 77, where he calls the geodesic equation a mere formal analogy to geometry.
A: I think one can enter a dispute regarding the notion of "accepted" but the idea is that General Relativity is successfully described by a Pseudo-Riemannian Manifold, subject to Einstein Equations, with free-falling objects following geodesics. Now you look for a set of axioms that give you this structure. One such set, although not entirely rigorous, is found in a paper by Ehlers, Pirani and Schild called the "The geometry of free fall and light propagation". I'll give you a brief discussion of the contents.
Start with two principles, (1) the Einstein Equivalence Principle and (2) the Finiteness of the velocity of light. The first one says that objects in free-fall in the gravitational field are in inertial movement and the second one says that not only light traves at a finite velocity but nothing goes faster than it.
Putting in another words principle (1) dictates that the trajectory of free-fall is as "straight" a line, from the point of view of an observer, as a constant velocity trajectory is in Newtonian physics. Principle (2) establishes that since everything travels at finite velocity there is a causal relation between events, namely if two events are have a spatial distance bigger than the velocity of light times the time separation they cannot have a cause-effect relationship, which of course implies in the relativity of simultaneity and other stuff from special relativity.
In more specific sense, principle (1) gives you a set of "straight lines", i.e. a set of geodesics, while principle (2) gives you a set of causal relations between events. In the paper by Ehlers, Pirani and Schild they call this two structures (1) a Projective Structure and (2) a Conformal Structure. Then they show that this two, with the assumptions that they are compatible and clocks behave in a reasonable way. imply the existence of a unique Lorentzian metric and Riemann tensor, together with the interpretation of geodesics and light-cones. All that's left if to require that the geodesics deviation be compatible with the one from Newtonian theory to obtain the Einstein Equations. 
They do provide a set of axioms that address every part of the assumptions, but it all can be traced back to these two principles, Einstein Equivalence and finite velocity of the propagation of light.
As a remark, you may note that this idea is very different from what Weinberg exposes, namely that geometry is not fundamental in this description, but, as he himself says in page 147, this is an heterodox point of view not subscribed to by general relativists. On the other hand it is, as far as I'm aware, the mainstream point of view in String theory.
