Why is "dark matter" theory accepted? Why wasn't general relativity rejected? Dark matter was made up to account for unexplained effects such as gravitational lensing, the speed of expansion of the universe, or the rate of rotation of certain galaxies.
However, as Feynman famously stated (see e.g. wikiquote),

If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is – if it disagrees with experiment it is wrong. That is all there is to it.

A theory has to stand the test of all observation. If there's an observation that contradicts the theory, science dictates that the theory must be rejected.
It's not the case with dark matter and general relativity. As it was discovered that the above-mentioned phenomena are not satisfactorily explained by general relativity and other modern theories, invisible "dark matter" was made up to patch the holes up. 
Why is it tolerated to grossly abuse the scientific process in this way? Yes, I'm aware that it would be really nice if it existed and everything would add up if it did. But is there really any evidence that it exists, other than that the numbers are wrong if it doesn't (which isn't really evidence at all)?
 A: 
A theory has to stand the test of all observation. If there's an observation that contradicts the theory, science dictates that the theory must be rejected.

"must be rejected" is not the correct conclusion. The theory as it stands is "falsified".
Falsification dictates  a "back to the drawing board" action. In the history of physics I know of no prominent rejection. Cogitations and study leads to the realization either that the range of the variables for which this mathematical theory is valid has to be limited or new inputs have to come that modify the predictions of the theory.
Newtonian Mechanics is a prime example. It was thought to apply to all ranges of energies and momenta and masses. Experiment showed its kinematic limitations and special relativity became applicable for the very small masses, and quantum mechanics became necessary for modeling data of molecules atoms and below in mass. Newtonian mechanics still rules in velocities of order meters/second and  masses of kilograms. Newtonian mechanics also became invalidated for the large masses of galaxies and clusters of galaxies and General Relativity was proposed  and validated by many observations. In their interfaces over the variable ranges all these theoretical models blend smoothly.
Dark matter was invoked even within Newtonian mechanics, i.e. for kinematic ranges where it was not really necessary to involve the mathematics of General Relativity. It is not a revolutionary modification in a physics model. The neutrino was proposed in order to keep energy and momentum conservation in beta decays, after all. And it does exist. The invocation of dark matter is on the same principle.
Experiments will show whether such weakly interacting particles exist and can contribute to the matter of the universe. If not, maybe new theoretical proposals will appear and be successful in describing the observations.
A: Let me begin by saying that although particular intellectual activities and particular theories are widely regarded as scientific, the "scientific method" or a demarcation criterion between science and pseudo-science isn't agreed upon universally. 
You raise an objection that scientists protected general relativity from falsification by data by inventing an auxiliary hypothesis, that of dark matter. The issue of auxiliary hypotheses is indeed controversial: if one can introduce a series of such changes to a "core" theory such that it evades falsification, is that core theory falsifiable? You go as far to say that the introduction of dark matter was an abuse of the scientific method.
This is a big topic and I don't think there is a definitive answer to whether an auxiliary hypothesis is "cheating" the scientific process or a legitimate response to new data. Imre Lakatos called the former a degenerative research program and the latter a progressive research program. Lakatos argued that if the modified core theory is better than alternative explanations, we should keep going with it, but if not, abandon it entirely. That sounds reasonable, if a little vague, and is arguably the case with general relativity and dark matter. Abandoning the core theory of general relativity would deprive us of many great explanations for observations.
I finish by adding that in my opinion, the issue of auxiliary hypotheses can be resolved rigorously with Bayesian confirmation theory. In BCT, one assigns numerical measures of degrees of belief to theories and updates them with data using Bayes' theorem. A succession of ad hoc and esoteric changes to a core theory leaves one with little prior belief in the theory (if there were many similar ways of altering the core theory), and might thus make it worse than an alternative theory. This is Lakatos' resolution to this problem, but in mathematics.
A: You are misrepresenting the scientific method.
Science (or at least Physics) procedes by formulating a mathematical model to describe reality then performing experiments to test the model. Human frailty being what it is this process is something of an ideal, but it's not a bad approximation.
As you say, the predictions of the pre-dark matter models didn't match reality, and that means the mathematical model is wrong. The response is to formulate a new improved model and test that. Two popular models emerged, dark matter and MOND. Of these dark matter dominates because it gives correct predictions in several unrelated areas e.g. galaxy rotation rates, galaxy cluster dynamics and gravitational lensing observations. MOND still has its adherents but making it generally covariant produces a model that looks a bit contrived to many of us. 
Though you didn't mention it, dark energy enjoys a similar status.
If you read the popular media they tend to talk about dark matter as if everybody just assumes it exists. While I suspect most of us think it probably does, you'll find few physicists (outside the lunatic fringe) who are willing to claim for certain that dark matter exists. Until we actually see dark matter directly (e.g at the LHC) there remains the possibility that some other effect is responsible for the phenomena we currently explain using dark matter.
So your criticism:

Why is it tolerated to grossly abuse the scientific process in this way?

is simply unfounded. The scientific method is working exactly as it should. We have a model employing dark matter that enjoys considerable indirect experimental evidence so our working hypothesis is that dark matter does exist. Should a more compelling model emerge we will waste no time testing that model instead.
A: "Why is “dark matter” theory accepted? Why wasn't general relativity rejected?"
The concept of dark matter has innocent beginnings. Literally, it just means something that doesn't emit light. When people first started trying to explain anomalies in terms of dark matter, they were thinking in terms of things like dead stars or lots of neutrinos, that could or should be out there, even according to a highly conservative physics. 
Nowadays the standard theory of dark matter is that there's some unknown particle, that doesn't interact much with anything, and which exists in enormous numbers in a spherical shell around every galaxy, and the gravitational influence of this dark matter "halo" is the cause of the anomalous galactic rotation curves. 
The simplest forms of that theory of dark matter, really are very simple. You may have heard of the standard model of particle physics, which is defined by an inventory of fundamental particles, their interactions and their other properties. That theory was built up in response to observation. Physicists had to deal with the fact that there's more to reality than proton, electron, electromagnetism, and gravity; in the end they had to add (among other things) quarks, neutrinos, and three "generations" of fundamental particles even though only the first generation is needed to make atoms. 
Given that new fundamental particles have repeatedly had to be introduced, it's really not a big step to just add one or two new types of particle to the standard model, that are inert with respect to the known forces except gravity, and which are produced in enormous quantities in the early universe. 
The simplest quantitative theories of dark matter are exactly like that, and apparently they fit the observations. So you especially can't fault such minimalist theories of dark matter, as being an abuse of the scientific method. They are highly conservative extensions of theories we have already been driven to adopt. The same can be said of field-theoretic models of dark energy, such as "quintessence". You observe something, accelerated expansion; you add one more fundamental field to your fundamental equation; and the observation is accounted for. 
It's true that, although these minimal models exist, and astrophysicists and cosmologists know about them and regularly test them against new data, the emphasis in the world of theory is on much more elaborate constructions. The dark matter particle may be just one of a multitude of "superparticles", the dark energy is a "vacuum energy" containing contributions from many fields, etc. 
These more elaborate constructions, like supersymmetry and string theory, are not driven by astronomical observation. They are driven by an attempt to distill the complexities of particle physics into something with fewer postulates than the standard model, and by attempts to solve the internal problems of theory, like the quantization of gravity. They typically predict that there are a lot of new physical phenomena not yet observed, and so these theorists do tune their models in order that some part of the model can explain the astronomical anomalies. But these paradigms are principally motivated by particle physics considerations, and so if your interest is mostly cosmological, they may leave you cold. 
The alternative approach that you favor - modify the theory of gravity - does also exist, in both the forms I mentioned (bottom-up minimalism and top-down high theory). MOND is the main minimal approach; and string theory can certainly provide a high-theory approach, e.g. by adding the dilaton to the graviton. Evidently theories of dark matter get the majority of the attention, but theories of modified gravity are also still being developed and studied. 
A real answer to your question would try to go through all the specific observations and the specific theoretical options, and explain why dark matter is more popular than modified gravity, but not decisively so. But there are many review papers that cover those topics, at a level detail that I can't. Instead I just wanted to sketch the overall theoretical situation so you can see its logic. 
