Gemara

(Mishnah): In the Mikdash, one [a Kohen] may return a bandage [on Shabbos, i.e. if he removed it to do Avodah];

This is forbidden outside the Mikdash.

In any place one may not put on a bandage for the first time.

(Beraisa): If a bandage fell off a wound, we may return it on Shabbos;

R. Yehudah says, [if it fell we may not return it, but] if it slipped below, one may push it back up, and vice-versa;

One may partially pull back a bandage to clean [part of] the wound, then pull back the other part to clean [the rest of] the wound;

One may not clean the bandage, for this is smearing. If he smeared, he is Chayav Chatas.

(Rav Yehudah): The Halachah follows R. Yehudah.

(Rav Chisda): They argue only about when it fell onto a Kli. If it fell on the ground, all forbid returning it.

Mar bar Rav Ashi: I saw a bandage fall from my father onto a pillow, and he returned it on Shabbos.

Mar bar Rav Ashi: Rav Chisda taught that the argument is when it fell onto a Kli, and Rav Yehudah said that the Halachah follows R. Yehudah [who forbids]!

Rav Ashi: I disagree.

Rishonim

Rif (35a): The Halachah follows the first Tana. Even though Rav Yehudah said that the Halachah does not follow him, Rav Ashi is Basra, and he acted like the first Tana. He holds that they argue even when it fell on the floor, and the Halachah follows the first Tana.

Question (Rosh 10:17): Why is the Rif so lenient when it fell on the ground? In the case of Rav Ashi, it fell on a Kli! Perhaps Rav Ashi meant 'I do not hold like Rav Yehudah, but I hold like Rav Chisda, that if it fell on the ground, all forbid'! The Halachah follows the first Tana, who permits when it fell onto a Kli. Alternatively, Rav Ashi argues with Rav Chisda. Rather, they argue about when it fell onto the ground. However, he holds like Rav Yehudah, who rules like R. Yehudah. Rashi explains like this.

Beis Yosef (OC 328 DH veha'Rosh): Surely, the Rosh had a mistaken text of the Rif, and it said that they argue also when it fell onto the ground. This is not in our texts of the Rif. Rather, he rules like the first Tana, since Rav Ashi acted like him. He can hold that this is only if it fell onto a Kli. The Rif explicitly says so in Shabbos (53b)!

Rashi and R. Yehonason (102b and 35a, DH Retiyah): The Mishnah discusses outside the Mikdash. The bandage fell, i.e. he did not intentionally remove it.

Ba'al ha'Ma'or: Rashi explains that the bandage fell by itself. One may return it, for this is not common, so Chachamim did not decree about this lest he smear. If he intentionally removed it, the Mishnah forbids. I hold like the Rif. In the Mishnah it fell onto the ground, and in the Beraisa it fell onto a Kli.

R. Yehonason (DH Machazirin): It is permitted because this is not common, so Chachamim did not decree. We forbid outside the Mikdash when he intentionally removed it. R. Yehudah forbids when it totally fell, but if it slipped, one pushes it above or below, like needed. All permit to partially expose it [and return it].

R. Yehonason (DH Lo): Falling on a Kli is like slipping [partially off the wound].

Bartenura (Mishnah 10:13 DH Kan): Rabanan forbid returning a bandage only if it fell onto the ground. If he removed it by hand, as long as it is in his hand, or even if he put it on a Kli, and it did not go onto the ground, one may return it anywhere.

Rambam (Hilchos Shabbos 21:27) If a bandage fell onto a Kli, one may return it. If it fell on the ground, one may not return it.

Rebuttal (Ra'avad): No, one may return it even if it fell onto the ground.

Perush ha'Mishnayos (10:13): Retiyah (bandage) is the cloth on which we put ointment to put it on the wound.

Rosh (ibid.): Rashi explains that one may not return a bandage lest he smear it. We are not concerned for grinding ingredients, since the bandage was on from before Shabbos. If we were concerned for grinding, why does the Mishnah discuss particularly a bandage? It should have taught all matters [of Refu'ah], even without concern for smearing!

Rosh (Avodah Zarah 2:10): One may do all needs of a Choleh through a Nochri. We similarly permit a Yisrael to do Shevus. Or, we permit only to avoid loss of a limb, but not for a Choleh without danger. One may not return a bandage outside the Mikdash. Or, perhaps that is not a case of illness, rather, mere pain, therefore we forbid Shevus with an act. This requires investigation.

Ran (Shabbos 53b DH Machlokes): The Halachah follows the first Tana. Therefore, it is permitted only if it fell on a Kli, but not if it fell onto the ground. The Rambam says so. The Ra'avad permits even if it fell onto the ground. It seems that he explains that Rav Ashi holds that they argue about when it fell onto the ground, and even so the Halachah follows the first Tana.

Poskim

Shulchan Aruch (OC 328:25): If a bandage fell from a wound onto the ground, one may not return it.

Beis Yosef (DH Kasuv): Shibolei ha'Leket (117) writes in the name of Ba'al ha'Itur "they argue about when it fell off unintentionally. If he removed it before Shabbos, or intentionally removed it on Shabbos, all forbid returning it." I do not know his source to distinguish Shogeg from Mezid.

Beis Yosef (DH v'Ha): Surely, we forbid returning for a Choleh without danger. If there is danger, one may put it on l'Chatchilah. One may put it on through a Nochri l'Chatchilah, for whatever has no danger, one may tell a Nochri to do it (Shabbos 129a).

Shulchan Aruch (ibid.): If it fell onto a Kli, one may return it.

Taz (19): The Tur brought from a Tosefta (Shabbos 6:3) that if a bandage separated from the knot, we return it under the knot, on the wound. The Beis Yosef (DH Isplanis) asked that since he permitted returning it when it fell onto a Kli, all the more so if it slipped! Therefore, the Poskim did not need to teach this. I say that one might have thought that one may not return it under the knot, lest he come to tie it if the knot will get ruined. We forbid due to tying. They decreed to forbid, just like when it fell onto the ground Chachamim decreed lest one smear. The words "under the knot" in the Tur are extra, to teach this [that we do not decree].

Magen Avraham (27): This is as if it slipped. If he removed it b'Mezid, he may not return it (Shibolei ha'Leket).

Machatzis ha'Shekel: The Beis Yosef did not know a source for Shibolei ha'Leket, but he does not disagree. Olas Shabbos (30) and Tosefes Shabbos (41) say that Rashi clearly holds like Shibolei ha'Leket.

Gra (DH Retiyah): This is an explicit Mishnah! One may not return a bandage outside the Mikdash. We do not distinguish how it fell [on the ground, or on a Kli]. R. Yehudah and the first Tana argue only because he removed it with his hands. See Rashi. Also the Ran (this should say R. Yehonason) explicitly say so there, unlike Ba'al ha'Ma'or and the Bartenura. How did the Beis Yosef overlook the Ran?!

Damesek Eliezer: The Mishnah discusses one who intentionally removed it, similar to the case in the Mikdash, in which he removed it in order to do Avodah. All forbid this [outside the Mikdash], even if he put it on a Kli. This is like the Ran, unlike the Ba'al ha'Ma'or and Bartenura, who explains that the Mishnah discusses [when it fell onto] the ground.

Magen Avraham (28): This is only if he is slightly sick.

Mishnah Berurah (82): He may not return it with a permanent knot, rather, with a bow. This is if it fell by itself. If he intentionally removed it, he may not return it. Some disagree, and permit. If he removed it to fix it, perhaps one may rely on them.

Shulchan Aruch (ibid.): One may put it on l'Chatchilah through a Nochri.

Taz (20): It is permitted through a Nochri if he is in great pain. If it is a slight pain, the Beis Yosef (DH Kasav) brought that the Ran (Shabbos 39a DH umi'Ha) forbids.

Mishnah Berurah (84): This is if his entire body is weak due to the wound, or there is danger to a limb. If he is only slightly sick, one may not do a Torah Melachah through a Nochri.

Kaf ha'Chayim (154): One may return it through a Nochri even if he intentionally removed it.