Gemara

(Rav Dimi): One who visits a sick person causes him to live. One who does not visit him causes him to die.

He means that one who does not visit him does not pray, not that he should live, nor that he should die.

(Ravin): Hash-m nourishes a Choleh - "Hash-m Yis'adenu (will sustain us) on the bed of the sick";

This verse also teaches that the Shechinah rests above the bed of the sick.

(Beraisa): One who visits the sick should not sit on a bed, bench or chair. Rather, he should cloak himself and sit on the floor, for the Shechinah rests above the bed of the sick.

Shabbos 12a (Beraisa): One who visits a Choleh on Shabbos says 'it is Shabbos, so we do not cry out (pray for a cure. In any case,) the cure will come soon.

R. Meir says, Shabbos can (cause Hash-m to) have mercy (and cure him)!

R. Yehudah says, (he says) 'Hashem will have mercy on you and all Cholei Yisrael.'

R. Yosi says, 'Hashem will have mercy on you amidst all Cholei Yisrael.'

Shevna of Yerushalayim says, when he enters (visits a Choleh on Shabbos) he says 'Shalom', when he leaves he says 'it is Shabbos, so we do not cry out. The cure will come soon, His mercy is great; rest peacefully.'

(R. Chanina): If there is a Choleh in one's house, he must request mercy for him amidst all Cholei Yisrael.'

This is like R. Yosi.

(Rabah bar bar Chanah): When we would go in back of R. Elazar to Mevaker a Choleh, sometimes he would say 'Hashem should bring Shalom upon you', sometimes he would say 'Rachmana (this is Arame'ic for 'the merciful one', i.e. Hash-m) should guide you to Shalom'.

Question: But R. Yochanan taught that if one prays for his needs in Arame'ic, the angels do not deal with his prayer (take it to Hash-m), for they do not understand Arame'ic!

Answer: A prayer said in front of a Choleh is different, for the Shechinah is with him:

(Rav Anan): The Shechinah is So'ed (sustains; some texts say So'er, i.e. Mevaker) a Choleh - "Hash-m Yis'adenu Al Eres Devai."

Support (Beraisa): One who is Mevaker a Choleh should not sit on a bed, for the Shechinah is at the head of a Choleh...

Rishonim

The Rif and Rosh (Shabbos 5b and 1:30) bring the Gemara in Shabbos verbatim.

Ran (DH Tanya): We may not cry out on Shabbos lest it arouse crying and anguish. R. Meir says that the merit of honoring Shabbos (by not feeling pained) can arouse mercy.

Rambam (Hilchos Shabbos 24:5): One who visits a Choleh on Shabbos says 'it is Shabbos, so we do not cry out. The cure will come soon. His mercy is great; rest peacefully.'

The Rif and Rosh (4:8) bring the hours not to visit a Choleh.

Ritva: In the last three hours, the Choleh looks so weak that a visitor will think that it is a vain prayer to request mercy for him.

Ran (Nedarim 40a DH Ein): In some situations one should pray that he die, e.g. he is in great pain and cannot recover. Rebbi's Shifchah prayed for him to die for this reason (Kesuvos 104a)

Note: Perhaps the Ran explains that one who visits in the last three hours is prone to despair of recovery, and will pray that he die!

Poskim

Shulchan Aruch (YD 335:5): If one prays in front of the Choleh, he may pray in any language. If he is not in front of him, he must pray in Lashon ha'Kodesh.

Taz (4): The Shulchan Aruch holds like the Rif (Berachos 7a), that angels understand only Lashon ha'Kodesh. The Rosh (Berachos 2:1) holds that they understand all languages, just they despise Arame'ic and do not deal with it. Why does the Tur follow the Rif against his father? Perhaps he is stringent to ensure that the prayer be heard, for a Choleh needs much mercy.

Tamim De'im (Sof 184, brought in Etz Yosef Sotah 32a - Hash-m heeds a prayer with great Kavanah in any language, for the gates of tears and Ona'ah (for one who was afflicted) are never locked.

Shulchan Aruch (6): We pray for the Choleh amidst all the sick of Yisrael - 'Hashem should have mercy on you amidst all Cholei Yisrael.' On Shabbos he says 'it is Shabbos, so we do not cry out. The cure will come soon.'

Beis Yosef (DH v'Yichlol and OC 287 DH v'Lo): Rashi explains that one must pray for the Choleh amidst all Cholei Yisrael in order that his Tefilah will be heard in the merit of the Rabim. The Ramban rules like R. Yosi. It seems that this is correct, for R. Chanina rules like him, but this is not the custom. The Tur rules like Shevna, and the Rambam rules like the first Tana. Perhaps they explain that R. Chanina holds like R. Yosi regarding including the Choleh with all Cholei Yisrael, but not regarding Shabbos. Since there is no indication of the Halachah regarding Shabbos, the Rambam rules like the first Tana. Perhaps the Tur agrees, and holds that Shevna explains the first Tana.

The Kotzker Rav (cited in Yosef Da'as, Shabbos 12a): When a person needs to and wants to cry out, but he cannot, this is the greatest cry. Therefore, the cure will come soon.

Shulchan Aruch (OC 287:1): One who visits the sick on Shabbos does not say like he does on a weekday, rather, 'it is Shabbos, so we do not pray. The cure will come soon. His mercy is great; rest peacefully.'

Taz (287:1): The Beis Yosef (above) said that Shevna explains the first Tana, i.e. he adds to his words. The Rambam does not mention 'His mercy is great...'; nor did the Shulchan Aruch in Yoreh De'ah. Why does the Shulchan mention it here?

Rema: Some say that he need not say 'His mercy is great...''

Kaf ha'Chayim (7): This is because this is a request, and we do not request on Shabbos.

Note: According to this, it should be forbidden to say this!