Do all of our discoveries of black holes in nature depend on the validity of GR? In the question Is there a black hole in the centre of the Milky Way? the answer by Motl seems to all but say the existence of that black hole is a fact (see also Evidence for black hole event horizons), whereas the answer by Bunn says "There's a strong consensus among astrophysicists". According to the Chandra X-ray Observatory FAQ the existence of that black hole is conditioned upon the validity of general relativity (GR); the answer there concludes "So, unless Einstein's theory of gravity breaks down, our galactic center must contain a black hole."
Questions are:


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*Does humankind's discovery of a black hole in the center of the Milky Way depend on the validity of GR? (If no, can you say why the Chandra X-ray Observatory FAQ contradicts that?) If yes:

*Do all of our discoveries of black holes in nature depend on the validity of GR? (If no, can you please elaborate on how one was discovered in a way that didn't depend on GR?)
Before answering please note: I'm not challenging GR in any way whatsoever here. I've tried to phrase the main questions so they can have unambiguous "yes" or "no" answers. As far as I can tell my questions aren't clearly answered elsewhere on this site. I realize that it would be difficult to discover a black hole without in any way depending on a theory of gravity, but I'm asking about a particular theory of gravity here, a theory that predicts black holes.
 A: The questions are ambiguous but let me answer:


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*Yes

*Yes


Summary: it's completely irrelevant
One could also claim that the answers to your questions are ambiguous because you haven't specified what part of GR the black hole discoveries should be dependent upon. Of course that they're dependent on GR at least to some extent because GR is our theory of gravity and gravity is essential both for the existence of black holes and their observation. One needs to use some knowledge about gravity and GR - with some possibly adjustable features - is the best theory we can use. One cannot interpret any observations without any theory. Ever.
If you ask whether the evidence for e.g. the black hole at the galactic center depends on the special features of GR that are not present in Newton's theory, well, that would be a more specific question but the answer would still be ambiguous. Of course that the existence of black holes depends on the difference of Einstein's gravity and Newton's gravity. In Newton's mechanics, there is no speed limit, so one can never create an object that would prevent light from escaping even in principle. One needs a relativistic theory of gravity to do so.
Someone might claim that this observation weakens some claims but it doesn't weaken anything at all because Newton's theory of gravity is perfectly ruled out (in the realm of high velocities or strong gravitational fields). Of course that only a relativistic theory is a viable candidate for a theory of gravity. So we are allowed to assume a relativistic theory of gravity and it doesn't weaken the conclusions, not even infinitesimally.
Also, it is fair to say that the main empirical evidence in favor of the galactic black hole has really nothing to do with any subtle features of general relativity. As discussed in the previous question, the main evidence is that energy is apparently being lost from our sight. It is not being returned by a corresponding thermal radiation at some point, and so on. More generally, the object is much less luminous than any object with an imaginable surface would be. This is not an argument depending on some subtle third-order corrections in GR; it is a very old-fashioned, down-to-earth, empirical argument.
There are two points you're not saying explicitly but you're saying them in between the lines as far as I can say - and they're the only reason I can imagine why someone would post this question in this form. The points of yours are


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*The evidence for the existence of black holes depends on details of GR - that's what you're asking about.

*The dependence on general relativity influences the robustness of the evidence.
The first point is strictly speaking tautologically right (we can't describe gravitational phenomena without any theory) but it is morally wrong because the main empirical evidence is of empirical character, indeed. But it's really the second assumption that is totally invalid and the very source of your problems. General relativity itself - or its key portions for this discussion - is even more well-established than the existence of black holes, so if the arguments that black holes exist depend on GR, it is not a problem at all.
A: I would rephrase your question as : what is the experimental signature of a black hole. If there exists a definite experimental signature of a celestial body that defines a black hole, your question is answered.
I found the following paper that clarifies the issue:

Classical black holes are solutions of the field equations of General Relativity. Many astronomical observations suggest that black holes really exist in nature. However, an unambiguous proof for their existence is still lacking. Neither event horizon nor intrinsic curvature singularity have been observed by means of astronomical techniques. This paper introduces to particular features of black holes. Then, we give a synopsis on current astronomical techniques to detect black holes. Further methods are outlined that will become important in the near future. For the first time, the zoo of black hole detection techniques is completely presented and classified into kinematical, spectro-relativistic, accretive, eruptive, obscurative, aberrative, temporal, and gravitational-wave induced verification methods. Principal and technical obstacles avoid undoubtfully proving black hole existence. We critically discuss alternatives to the black hole. However, classical rotating Kerr black holes are still the best theoretical model to explain astronomical observations. 

From this it is evident that in contrast to most physics subjects where first there is experimental evidence needing explanation and then theory arrives, black holes are an "artefact" of general relativity theory, i.e. the concept carries all the baggage of GR.
Nevertheless, what is called a "black hole" in GR has some experimental signatures which any other competing gravitational theory would have to account for. It might not be in the enticing format of a "black hole", but some data are there. These at the moment are consistent with the theoretical black hole expected from GR.
So in a sense your question has as answer : "yes the definition of a black hole is within the terminology introduced by general relativity" and may not be defined outside it;
but also
"no the experimental signatures do not depend on general relativity in order to exist, just their interpretation and attribution as a black hole" may be in question,  if an alternative theory to GR succeeds to describe reality.
