The questions are ambiguous but let me answer:
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
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.