Can General Relativity be correct if Special Relativity is incorrect?
No, special relativity is a special domain of general relativity, hence the name. Special relativity is the limit of general relativity when gravitational effects become negligible. You can't have a theory that's completely correct but has a wrong limit (I use limit here not in the strict mathematical sense, but as like a "domain").
No, SR (Special Relativity) is an intrinsic essential component of GR (General Relativity).
GR says that in perfectly flat spacetime mechanics behaves exactly according to the rules of SR as the answer of enumaris states.
GR also says that stress-energy causes spacetime curvature, but if you chop a region of curved spacetime into sufficiently small subregions then the curvature in each subregion becomes negligible and we can then use SR inside the subregion. To be more precise, as the size of a subregion approaches zero, so does its curvature, and so the rules of SR are valid locally.
This is similar to how we can chop the curved surface of the Earth into a set of flat maps which we can put into an atlas. Each page of the atlas tells us what's in that region, what the scale of that page is, and how that page connects to the other pages in the atlas.
So, roughly speaking, GR tells us how to construct an atlas of curved spacetime so that we can use SR locally on any atlas page. GR gives us a lot of freedom in how we chop up spacetime, and in what coordinate systems we use for our atlas pages.
(Unfortunately, this gets very difficult to do exactly when the spacetime curvature isn't simple or symmetrical: even the general two body problem in GR has no analytic solution).
TL;DR: GR says that SR is always valid locally, but in regions of high curvature the locale may be very small.
From a theoretical point of view, one may argue that it depends on how one defines the words GR & SR, and how abstract one wants to be. For starters one may generalize to other than 4 spacetime dimensions. Moreover, the local spacetime model could in principle be different from Minkowski spacetime. Instead of Einstein gravity, one may consider e.g. conformal gravity, SUGRA, Newton-Cartan gravity, etc.
As far as I know, and I'm not an expert or a physicist, Special Relativity developed the principle of Relativity which stipulates mainly two rules:
- All events follow the same laws of physics in every frame of reference.
- The independence of the speed of light, which means that light is always moving at a constant value (in the vacuum, at 299792458 m/s), even if one is in an inertial frame and another accelerating.
General Relativity also follows this principle, because the speed of light is always the same in vacuum for all observers, it doesn't change. This might be incorrect, but I think that also GR without SR it would not work. We can take the ISS as an example: We know that the gravity becomes weaker the further that you go from the source of the gravitational field; in our case, the ISS is further from the earth, our source of gravity, than us. Because of the time dilation due to gravity, the further that you go from the source, the faster that time passes, so we might guess that in the ISS the time for the astronauts goes faster than for us. But we need to remember that the ISS is moving, and is moving so fast, fast enough for the relativity effects to be non-negligible, and we know that the fastest and object is moving, the time dilation increase due to the Lorentz factor, and the time gets slower. So having this two assumptions, the SR effects are bigger in the ISS than the GR effects, so the time there gets slower than for us, and the astronauts in the ISS are getting older slowly.
I hope that this could help you.