How much of general relativity does the equivalence principle actually imply, why is it different? From EP we have that gravity is not a force but a pseudo force, i.e
an inertial force due to a gravitational field an accelerating a body 
independantly of its mass, in other words, the trajectory of any body only depends on its initial position and velocity, but not on its composition/mass.


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*Pseudo forces are well described by newtonian mechanics in the accelerated (non-inertial) frames formalism, so why not gravity? Pseudo forces are taken into account by a change of coordinates, as for example fictitious forces on earth are just the acceleration terms coming from the curvature of spherical coordinates. Therefore these pseudo forces could also be described by 
a metric and a connection and everything in a manifold...


But then why do we need to describe spacetime as a curved manifold and geometrizing (curvature,metric...) gravity?
Can't we stick with the (old newtonian gravity+SR) approach, what makes it false?


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*Is it that this old approach does not fulfill SR? But then SR describes dynamics, so can't we describe gravity as a force in SR (it doesn't work?)?
Is it because a newtonian grav. field acts instantaneously (faster than speed of light)?
In the end, my question is:


How much of GR formalism is derived/constrained from the EP alone? 
What's different from (newtonian gravity(contains EP)+SR), why is it better?
Or are curvature+metric+connection vs Newton gravity (fictitious force)+SR two ways to describe the same physics?
 A: GR does not have to be described in a formalism of curved spacetime. You can have, e.g., Deser's spin-2 field on flat spacetime. This theory is inconsistent until you add corrections, which end up making it equivalent to GR.
The equivalence principle is what prevents us from deciding whether a theory like Deser's is what "really" happens, as opposed to the standard formalism of GR using differential geometry. Ultimately the only way we have of telling whether a ruler is straight is to send a test particle along it. There is no way to tell whether the trajectories of test particles are "really" straight or "really" curved.
Misner, Thorne, and Wheeler has a nice discussion of this sort of thing in ch. 17. They discuss distinguishing features of theories of gravity that include the EP, prior geometry, and the existence of auxiliary fields.

Is it because a newtonian grav. field acts instantaneously (faster than speed of light)?

Yes, this certainly forces us to make a field theory rather than a theory of instantaneous action at a distance.
