Gemara

(Mishnah): Chachamim forbid eating cheese of Nochrim, but permit benefit from it.

(R. Yehoshua): It is because they use stomachs of Neveilos to curdle it. (Note: 'stomach' refers to what is inside it, unless we specify the skin.)

Objection (R. Yishmael): The Isur of an Olah is more severe, yet the Torah permits a Kohen to suck the stomach of an Olah!

35a (R. Shimon ben Pazi): Perhaps a snake drank from the milk.

(Shmuel): They curdle the milk in the skin of stomachs of Neveilos.

Rishonim

Rambam (Hilchos Ma'achalos Asuros 3:13): Chachamim of the Mishnah forbade cheese of Nochrim, because they curdle with skin of stomachs of Neveilos. The skin is very small compared to the milk. It is not Batel because it is Ma'amid (curdles), therefore, all the cheese is forbidden.

Rambam (14): Cheese that Nochrim curdle through herbs or fruit juice, e.g. fig sap, and one tastes them in the cheese, some Ge'onim forbid it, because Chachamim already forbade all cheese of Nochrim, whether it was curdled with Isur or Heter. We forbid the latter due to what was curdled with Isur.

Ra'avad (13): The Gemara gave other reasons to forbid! The Ge'onim forbade according to letter of the law. It was not a mere decree.

Beis Yosef (YD 115 DH Gevinos): The Rambam said this in the name of some Ge'onim, but it seems that he holds like them, for he did not bring an opposing opinion! This refutes people of Italy who permit cheese of Nochrim, because they curdle them with flowers. R. Tam rules like R. Yehoshua ben Levi who forbids due to Giluy. Even though nowadays we do not forbid exposed liquids, the Isur of cheese of Nochrim stands. Semak explicitly said so according to R. Tam. Even though Tosfos is no stricter about cheese than water regarding Giluy, perhaps they did not say so to be lenient in practice. Tosfos' first words, 'R. Tam says that nowadays there is not a simple reason to forbid', support this. Tosfos proceeds to say that in many places people eat cheese of Nochrim because they curdle with flowers, and that Chachmei Narvona permitted in their place. This does not suffice to permit. Those Ge'onim (of Narvona) ruled like the opinion that forbids due to the skin of stomachs of Neveilos. However, R. Tam holds that it is due to Giluy, so we can say that even nowadays it is forbidden. All the more so, since the Rambam and Rashba, pillars of the world, forbid, and in all other places we have heard, Yisraelim forbid whether it was curdled with herbs or with stomach, one may not separate from Klal Yisrael and breach the fence of Yisrael Kedoshim that Chachamim of the Mishnah fenced.

Hagahos Ashri (Avodah Zarah 2:26): One may curdle cheese in the stomach of a Neveilah or of a Kosher animal that nursed from a Tereifah, and salt (what is in) the stomach in its skin, like the first words of Rashi. However, R. Tam forbade l'Chatchilah to use stomachs of Neveilos, for this is repulsive. B'Di'eved, he permitted eating it. However, in all their places in Eretz Kena'an the custom is to forbid cheese of Nochrim and in stomachs of Neveilos. Chas v'Shalom for one with Yir'as Shamayim to permit this! R. Simchah said that the custom is to forbid intestines in which milk was found, even if they were not salted with the milk, for they curdle it and sour it.

Hagahos Ashri (40): If a Yisrael went to a village to buy cheese, and they did not know that the Yisrael intends to harden all the milk, even if he did not see the milking, it is permitted. (If they wanted to mix in Tamei milk, surely they would mix it with the milk that they would curdle - PF.) Even if you want to forbid, if he saw everything except for the start of the milking, it is permitted. Once, a Nochri told a Yisrael 'buy the cheese (that I will make)', and the milk was already milked, and he made it, and R. Tam permitted. However, one should permit only to prevent a big loss.

Beis Yosef (YD 115 DH v'Chasuv b'Hagahos): A Yisrael had goats with a Nochri. Every morning he sent his daughter to see, and most or all of the milking was finished before she arrived. Semag says that R. Tam was upset with. Semag said that R. Chananel rules like R. Yehoshua ben Levi, and even though we are not concerned for Giluy nowadays, we may not be lenient about cheese of Nochrim, for it was decreed through a vote. Even though the reason does not apply nowadays, a new vote is needed to permit it. Semak says in the name of R. Tam that we permit exposed water, and we do not decree due to places where there are snakes, because there is no other reason to forbid, so where there are snakes, people will not come to be lenient. The Gemara gave many reasons to forbid cheese of Nochrim, e.g. the skin of stomachs of Neveilos, so if we permit in places where there are no snakes, people in places where there are snakes might come to permit.

Taz (10): I.e. if people find that cheese of Nochrim is permitted in a place, they will think that the Isur was not due to Giluy, rather, to another reason, and they will permit even where there is concern for Giluy.

Poskim

Shulchan Aruch (YD 115:2): Cheese of Nochrim is forbidden because they curdle in the skin of the stomach of Neveilos. Even if they curdle with grass, it is forbidden.

Beis Yosef (DH Gevinos): In many places people eat cheese of Nochrim, for they curdle with flowers. This is why Chachamim of Narvona permitted in their place. However, R. Yitzchak says that here they curdle with stomach, so there is reason to forbid, for they salt the stomach in its skin. The Mordechai (826) says that R. Tam permitted only when he milked to make cheese, but not of he milked for (immediate) consumption or Kutach (a dip made with milk and stale bread).

Darchei Moshe (2): The Mordechai in Chulin (740) says that our custom is not to buy it even if a Yisrael saw from the milking until cheese was made. The Agudah says that Gedolim permitted one who buys a number of cheeses from a Nochri, and the Yisrael sees the Nochri make them on Shabbos and does not speak, but this is repulsive.

Shach (20,21): The Agudah agrees with the Mordechai. The Agudah wrote this in Maseches Shabbos, regarding Hilchos Shabbos. He could agree that cheese of Nochrim is forbidden! He permits buying the cheese (made) on Shabbos, in order to sell it to a Nochri, or in places where the custom is to permit cheese of Nochrim like R. Tam. The Yisrael must supervise that they do not switch cheeses. Alternatively, the Agudah's case is different, for the Yisrael already bought the cheese and it is his, so it is not considered cheese of Nochrim. Surely it does not help for a Yisrael to see the cheese made, for Chachamim forbade all cheese of Nochrim, even if they curdle with Heter, for they did not distinguish. Regarding milk, we distinguish. Teshuvos Maimoniyos (8) says that R. Shimshon forbids even if the Yisrael saw from the milking until the cheese was made. Rabbeinu Meir permits only if the Yisrael put the stomach into the milk. Hagahos Ashri, Isur v'Heter and R. Meir permit when we know that the Nochri intended to make cheese. I.e. they permit a Yisrael to make cheese from the milk. The Rema permits even if the Nochri made the cheese. This is wrong. The custom is to buy cheese from vendors, even though Nochrim make it, because Yisre'elim rent the animals, so the (milk and) cheese belongs to Yisre'elim. Chachamim never forbade cheese of Yisrael that Nochrim made. The custom is that a Nochri in a Yisrael's house makes cheese for the Yisrael. Even if they do not rent the animals, if the Yisrael buys the cheese before it is made, it is permitted. If the Nochri owned it while it was made, even if a Yisrael was there, it is forbidden unless the Yisrael did an action.

Rema: This is the custom, and one may not breach it, if not in a place where it has been the custom to permit it from of old. If a Yisrael sees the making of the cheese and the milking, it is permitted.

Gra (14, from Maharam 374): A Tosefta teaches about bread and cheese that a Nochri and a Yisrael made. Just like the Yisrael's presence does not permit bread unless the Yisrael did one of the primary Avodos, the cheese is permitted only if the Yisrael put the stomach in the milk.

Rema (ibid.): This is the custom in all these lands. If one saw the making of the cheese but not the milking, one may permit b'Di'eved if there is no concern lest he mixed in something Tamei after making the cheese from the milk, for something Tamei does not curdle, so surely the Nochri did not mix it in, since he intends to make cheese. In any case, one may not eat the milk like this.

Taz (11): R. Tam permitted if one saw the milking. Why does the Rema permit even without this milking? The Shulchan Aruch rules like the Rambam who does not distinguish in this case, and forbids even if they curdled with herbs! Also, if we permit, people will come to drink the milk! The Rashba forbids a piece proper (to honor guests with) lest it be confused with an improper piece, even though the Isur is mid'Rabanan. All the more so, if we permit the cheese, we should be concerned lest people permit the milk, which could contain Tamei milk, which is an Isur mid'Oraisa! I say that we should permit only if a Yisrael saw the milking, especially since according to R. Tam we should forbid in any case!

Gra (15): According to Tosfos, if a Yisrael saw the curdling, this removes all concerns except for Giluy, which nowadays is not a concern.

Shach (22): Hagahos Ashri permits only to avoid a big loss. However, if a Yisrael saw the milking, but was not there from the beginning, he permits even without a big loss.

Shach (25): If we know that the Nochri intended to make cheese, it is permitted even if he has a Tamei animal in his herd.