On shabbos there is a rabbinic prohibition against doing many medicinal things. Is the whole requirement that he is doing it for medicine and that it looks medicinal, or must it actually be scientifically medicinal? The difference would be: is homeopathic or other medicine that science says do not work any more than a placebo, prohibited on shabbos?
The prohibition of taking medicine for someone who is only slightly ill comes for the gezeiro/decree of shehikas sammonim (grinding materials to make the medicine) as it says in Pninei Halocho
הוסיפו חכמים וגזרו על מי שסובל ממיחוש או מקצת חולי שלא יעסוק כלל ברפואות, היינו שלא יאכל או ישתה תרופות, ולא יסוך את גופו בתרופות, ולא יעשה פעולות שנועדו לרפואה, משום שחששו שמא מתוך שיהיה טרוד להסיר את המיחוש, יבוא לשחוק סממנים (לטחון צמחים לאבקה) כדי להכין תרופה, ויעבור על איסור תורה של 'טוחן'(שבת נד, ב, ורש"י
Chachomim added a decree on someone who is only slightly ill not to be busy in any way with healing processes, that is not to eat or drink medicines or to anoint himself with them or do anything that is linked to a cure, because they were concerned that he would be so worried by his condition that he would come to grind the medicines and transgress the issur of grinding.
Homeopathic medicines although they are very very highly diluted, do normally start off with some active substance. See here under “Preparations and treatment”. Therefore if the person believes that a homeopathic medicine will help him, he may come to the issur and therefore the decree/gezeiro should remain in place.
R Meir (Shabbos 67a) permits wearing a fox's tooth on Shabbos as a sleep aid. The Rambam (Hil' Shabbos 19:13) rules like Rav Meir.
Chullin 77b discusses whether such "cures" would qualify as adopting foreign customs, derekh Emori, and thus prohibited. The Ran (ad loc) explains that anything that healed three times is considered a verified cure. (A statement we find Chazal make about amulet writers on Shabbos 61b.) We do not need to have an explanatory theory.
The Rambam says similar in the Guide to the Perplexed 3:37. I normally would not assume something written in the Guide is halachic, but since it matches the Ran's position, I think it's the most likely explanation of his ruling in his code.
One bit about the Rambam, though, he permits the cures because "they were thought to be effective." In other words, placebo effect would be considered sufficient reason to permit a cure. (See also his commentary on mishnah, Yuma 8:6.)
So, you end up with two questions:
1- Now that science has developed more certain validation of therapies, would Chazal still use the "worked three times" criterion? In other words, is this criterion an actual halachic statement, or were they just giving the best validation system then known?
This question is actually still under dispute. (See Nishmas Avraham vol 4 OC 306 who follows R SZ Aurbach requiring double-blind testing, but he does cite both sides.)
2- A more pragmatic question: If we do still use the "worked three times" criterion, does the "treatment" in question actually qualify? (I am taking it for granted that if it actually passed a double-blind statistical test, it wouldn't still be labeled "alternative medicine".)