Why is homeopathy not acceptable as scientific medical healing system?
Because it has never passed a properly controlled, double blind trial. We normally don't accept things as "pure scientific" without tests.
A scientific test of a claim is fundamentally a means of observing, documenting and measuring the claimed phenomenon while eliminating all other explanations (or at least as many as possible) for it.
First homeopathy has a lot of anecdotal evidence supporting it, it does not pass rigorous replicable scientific tests. As long as there are a number of interfering factors (placebo effect, experimenter bias, etc.), homeopathy "works", but the moment these other explanations are eliminated (double-blind, randomized, placebo-controlled trials), the effect disappears.
Second, because the body is so good at healing itself, it is remarkably easy for any physician to convince themselves that a useless intervention is having a beneficial effect. This is why evidence-based medicine was developed. Objective testing of treatments to see whether the effect is real or coincidental/illusory have led to many cherished but useless therapies being withdrawn.
Third, the mystery thing about homoeopathy is the remarkable tenacity of its adherents in clinging to the belief that there's a real effect going on. Consider the claim is that the effect is so striking that it is obvious even at the level of the clinical anecdote, and indeed many positively miraculous cures are reported. Any effect this strong should be very simple to demonstrate by a controlled trial. Testing homeopathy the best anyone can say is there might be an effect there, but then again there might not. That does not sound like a good basis to be claiming to heal people.
Fourth, homeopathy depends entirely on coincidental recovery and wishful thinking for its effects, and those who believe that the remedies are actually doing something are simply mistaken. Nobody will ever win the Nobel Prize for trying to prove there is science behind a delusion. The history of the Pons/Fleischmann experiment in cold fusion as an example.
There is good reason why homeopathy is not acceptable as a science.