In the case of somebody who needed an injection on Shabbat, and after the injection he wants to use a cotton ball to clean the area where the injection was made, a number of Halachic questions arise. Firstly, if he does not have cotton balls, and needs instead to rip a piece of cotton from a cotton roll, the question becomes whether this violates the prohibition of Kore’a – tearing on Shabbat. Indeed, Hacham Bension Abba Shaul (Israel, 1924-1998), in Or Le’sion (36:22), writes that it is forbidden to rip cotton on Shabbat, for this reason.
Hacham Ovadia Yosef, however, disagrees. In Hazon Ovadia – Shabbat (vol. 3, p. 377), he writes that cotton does not qualify as a "Hibur Gamur" – meaning, the strands of cotton are not completely attached such that removing a piece would constitute "tearing" with respect to the prohibition of Kore’a. According to Hacham Ovadia, then, it is entirely permissible to tear a piece of cotton on Shabbat.
A second question arises regarding the use of cotton to rub alcohol on the wound. The Halachic authorities note that if one pours alcohol onto a piece of cotton, it is all but certain that he will end up extracting absorbed alcohol from the cotton, in violation of the prohibition of Sehita. Cotton is so porous that it is virtually impossible to avoid Sehita when handling cotton after pouring liquid on it. Therefore, one would either have to use a material other than cotton for applying the alcohol onto the skin, or pour the alcohol directly onto the skin and then gently pat the moist skin with cotton. This is the ruling of Rav Moshe Ha’levi (Israel, 1961-2000), in his Menuhat Ahaba.
Summary: It is permissible to rip a piece of cotton from a cotton roll on Shabbat. One may not, however, pour alcohol onto a piece of cotton on Shabbat, as he will then inevitably end up extracting absorbed liquid from the cotton. Therefore, one who needs to apply alcohol to his skin on Shabbat must either use a different material or pour the alcohol directly onto the skin.