The Shulhan Aruch (Orah Haim 287) rules that is permissible to visit sick patients and comfort mourners on Shabbat. He adds, however, that one should formulate his wish to the ill patient differently on Shabbat than he does during the week. Rather than directly wishing him a full recovery, one should say, "It is Shabbat, and thus we must not cry out, but healing shall soon arrive" ("Shabbat Hi Mi’liz’ok U’r’fu’a Keroba La’bo").

The Mishna Berura (commentary by Rav Yisrael Meir Kagan, 1839-1933) writes that one should not specifically schedule a visit to a sick patient or mourner on Shabbat. The Gemara says that the Sages permitted visiting patients and mourners on Shabbat "with great difficulty" ("Be’koshi"). Such visits cause one distress, which we must of course try to avoid on Shabbat, and thus in principle, these visits are problematic on Shabbat. The Sages nevertheless granted permission for those who cannot visit patients or mourners during the week to do so on Shabbat. It is improper, however, to unnecessarily plan such a visit specifically for Shabbat.

As mentioned earlier, Halacha requires a person visiting a sick patient on Shabbat to say that "healing shall soon arrive" without actually praying on the patient’s behalf, which is inappropriate on Shabbat. The implication of this Halacha is that prayer for ill patients is unnecessary on Shabbat, because Shabbat itself has the capacity to restore health. One Rabbi commented that this notion is alluded to in the verse, "Rak Shibto Yiten Ve’rapo Yerapeh" ("He shall only pay for his lost work, and for his medical costs" – Shemot 21:19). The Torah teaches that "Rak Shibto Yiten" – one needs only the Shabbat, and then "Ve’rapo Yerapeh" – he will be cured. This is also a reason why the Torah requires circumcising a newborn infant specifically on the eighth day, which ensures that the child will have experienced a Shabbat before his Berit Mila. The healing powers of Shabbat strengthen the newborn so he is healthy enough to endure the circumcision. This principle also underlies the Gemara’s comment, "All foods – a person may eat them for health purposes on Shabbat" ("Kol Ha’ochalin Ochel Adam Li’r’fu’a Be’Shabbat"). The plain meaning of this passage is that a healthy person may eat food for health purposes on Shabbat. Additionally, however, it means that all foods that a person eats on Shabbat have the effect of Refu’a (healing), because Shabbat observance has therapeutic powers. Therefore, we do not need to pray on behalf of ill patients on Shabbat, as the Shabbat observance itself has a healing effect.

Summary: It is permissible to visit ill patients and mourners on Shabbat, though one should not specifically schedule such a visit on Shabbat. It is only if he unable to visit the individual during the week that the Sages permitted paying the visit on Shabbat.