Gemara

(Beraisa): If a Yisrael desecrates Shabbos in private, but not in public, he can be Mevatel his premises. If he desecrates Shabbos in public, he cannot be Mevatel his premises. This is because Chachamim said that (only) a Yisrael can give permission or be Mevatel his premises.

Chulin 4b - Suggestion: If Yehoshafat drank Achav's wine, he must hold that a Mumar to idolatry is not like a Mumar to the entire Torah!

Rejection: He is a Mumar to the entire Torah (he is considered like a Nochri). Chachamim had not yet decreed against wine of Nochrim.

Eruvin 69a: David went out (on Shabbos) with a band of fragrances. When he saw R. Yehudah Nesi'ah, he covered it. R. Yehudah Nesi'ah said that David can be Mevatel Reshus according to R. Yehudah. (He is not like a Nochri.)

Rishonim

Rif (Eruvin 21a): One who is Mechalel Shabbos in the market is like a Nochri, so he cannot be Mevatel Reshus.

Hagahos Ashri (Chulin 1:7): After he transgressed (idolatry or Shabbos) two or three times, one may not eat from his Shechitah, for he is like a Nochri. The first time he was Shochet (to idolatry or on Shabbos) is not forbidden. There is no clear proof to permit what he slaughters after this.

Chidushei ha'Ran (Chulin 4b DH Mai): Stam wine of Nochrim is not Asur due to concern for Nisuch, rather, due to intermarriage. If a Mumar to idolatry is concerned a Mumar to the entire Torah, he is included in the decree, even though the Torah does not forbid marrying his daughter.

Rivash (4): One who is Mechalel Shabbos in public is a Nochri in every way. Wine that he touches is Asur.

Poskim

Beis Yosef (119 DH Od, citing R. Yonah, and DH Kasav ha'Rashba): One who is Meshumad to be Mechalel Shabbos in public, or does not believe in Divrei Chachamim, he is a Min, and his wine is Yayin Nesech. This is if he was established to transgress three times, in front of 10. The Rashba (Teshuvah 430, 687) says that the former is a Meshumad in every way.

Nekudas ha'Kesef (YD 124:1): The Mabit (2:38) forbids wine of Kara'im who are Mechalel Shabbos, like that of Nochrim. This connotes that even touching forbids, like the Beis Yosef brings from the Rashba.

Chasam Sofer (2:120, cited in Pischei Teshuvah (1): Levi pressed wine, and two witnesses testified that he did so on Shabbos night after Tzeis ha'Kochavim, and two testify that it was not yet Tzeis ha'Kochavim. The latter witnesses agree that it was very close to Shabbos, and he is a Rasha. He should be rebuked, and one may not believe him about Isurim. However, his wine is not Yayin Nesech. He is called Mechalel Shabbos in public only if he did so in front of 10 Yisre'elim. The two pairs of witnesses contradict the other, so we leave him on his Chazakah. Also, perhaps they do not contradict each other. One pair saw three scattered stars, and the other pair did not see them. He relied on the latter pair (that it is not yet Shabbos). Also, they did not see that the grapes needed to be minced and crushed. Horayos 11a discusses a Mumar l'Te'avon for an Aveirah mid'Rabanan (Tana'im argue about whether he is a Mumar). The reason that a Mechalel Shabbos makes Yayin Nesech is not due to intermarriage. One may marry his daughter! It is not due to idolatry. It is a mere fine to consider him to be a Nochri who serves idolatry. Therefore, we need not be so strict about it, especially if we do not know that he touched the wine, just that he stomped through his Ko'ach. If so, the wine that came out before he became a Mumar is permitted. What came out afterwards in the same winepress mixed with the first wine, and every drop is Batel in Heter when it mixes in. Therefore, we should not be stringent to forbid the wine of a Kosher Yisrael due to this.

Mishnah Berurah (385:6): If one is ashamed to be Mechalel Shabbos in front of a great person, even though he transgresses in front of many, this is called he transgresses in private.

Note: Some rely on this only for Eruvin (R. Y. Zilberstein). Some are stringent about Yayin Nesech (Teshuvos v'Hanhagos 2:400). Some are lenient only if the Chilul Shabbos was mid'Rabanan (Darchei Teshuvah 119(34)), and if he keeps most Mitzvos (Shuvah Elai p.12-15 brings all these opinions).

The questioner in Har Tzvi (YD 105 DH Mechalel): The masses say that wine of a Mechalel Shabbos in public is Yayin Nesech, but no Posek says so. Nisuch depends on idolatry, not on Chilul Shabbos. The Beis Yosef says so in the name of R. Yonah, and says that the Rashba says that it is only after three times. This shows that it is not the Halachah, for there is no reason to distinguish between two and three, like Tosfos (Chulin 14a DH ha'Shochet) says. The Rashba said so in the way of Musar, but not that this is the Halachah.

Rebuttal (Har Tzvi): Some Rishonim argue with Tosfos. We can say that the Rashba is among them. Regarding slaughter to idolatry, the Rashba holds that one becomes a Mumar after one Aveirah. The Shach (YD 4:4) asked this contradiction and did not resolve it. Tevu'os Shor (11) says that the Rashba is more lenient about Yayin Nesech. Neither said that the Rashba does not truly forbid Yayin Nesech! Nekudas ha'Kesef proves from the Rashba that even one who does not serve idolatry makes Yayin Nesech. Also Hagahos Ashri connotes that one must transgress three times to become a Mumar. He is unsure about one who transgressed less than three times. Tosfos merely said that once does not make a Mumar. Perhaps also he is unsure if three times are needed! Bahag explicitly says that a Mechalel Shabbos in public is like an idolater, and he makes Yayin Nesech. No one explicitly argues. However, perhaps Bahag means that if a Mechalel Shabbos says 'I pour this wine to idolatry', we believe him. We do not say that he merely intends to vex the owner. He did not discuss wine that he just touches. However, it seems that he forbids even what he touched. Ha'Eshkol understood this way, and says that he is no better than a Nochri who does not serve idolatry, and one may not drink wine that he touched, but one may benefit from it. The Ran says that if a Mumar is considered a Nochri, the decree of their wine applies to him, even if the reason (intermarriage) does not apply. It is difficult to distinguish between a Mumar to idolatry and a Mechalel Shabbos in public. An egg laid on Yom Tov is forbidden. Some say that it was included in a decree about fruits that fell off a tree on Shabbos or Yom Tov, and some say that it is included in a decree lest one squeeze Peros (Reish Maseches Beitzah). Even though these decrees do not apply to eggs, eggs were included. R. Akiva Eiger (Teshuvah 5) says that the decree does not apply to milk milked on Shabbos or Yom Tov, for the reasons do not apply. We include only what we find that Chachamim included. However, the Shulchan Aruch calls a Mechalel Shabbos in public a full Nochri. It did not need to say that his wine is forbidden.

HarTzvi (DH u'Mah): The Mechaber (YD 2:6) brings the Rambam's opinion that one must check the knife of a Mumar to one matter, but not of one Pasul for testimony. The Shach (19) explains that a Mumar is one who is Mu'ad to transgress (Pri Megadim - he transgressed three times), and a Pasul is one who transgressed only once. We kill a Moser only if he is Muchzak to do so (CM 388:1). The Shach (388:57) cited Teshuvas Maharam bar Baruch that this is after three times. The Poskim argue about whether or not disqualifying a person is considered a capital case. (If it is, we require three times, like R. Shimon ben Gamliel. If not, twice suffices, like Rebbi.) Hagahos Ashri was unsure about this.

Igros Moshe (OC 5:37:8): One may not give wine to a Yisrael Mumar idolater, for he may not drink it. Does the same apply to a Mechalel Shabbos in public? The primary decree was only for real Nochrim due to intermarriage. We find that it applies even to a Yisrael Mumar to idolatry (Chulin 4b). After the decree, Yehoshafat could not drink Achav's wine. However, if a Yisrael Mechalel Shabbos in public does not serve idolatry, there is no concern for Nisuch. We do not find in the Poskim that he makes Yayin Nesech. The Beis Yosef brought this from the Rashba, and Nekudas ha'Kesef brought it, but they did not put it in the Shulchan Aruch and Shach. It was not so clear to them. The custom is not to drink wine that he touched. A Sefer attributed to Rashi says so. We may decide Halachah based on a custom to be lenient, but not based on a custom to be stringent, for anyone may be stringent if he is unsure. Also, perhaps it is not a universal custom, rather, only for Yir'ei Shamayim. Since the Isur is not clear, and it seems that it is only a custom and not letter of the law, we are not stringent not to give to a Mechalel Shabbos to drink. Even if this is due to animosity, this is the custom. Even if there is a Safek that perhaps some are stringent about this, we are lenient about a Safek mid'Rabanan.

Tzitz Eliezer (8:17): Minchas Pitim (YD 151) permits selling Kosher wine to a Mumar, based on the Shach who says that there is no Isur to help a Mumar sin, unless he could not sin by himself. Dagul me'Revavah says so about anyone who wants to sin b'Mezid. Often, it is not clear if one is Mechalel Shabbos in public. Many Rishonim and Acharonim are lenient about Lifnei Iver whenever there is any Safek. Minchas Pitim permits to sell even wine of Nochrim to a Mumar, for in any case the buyer forbids it.

Tzitz Eliezer (12:56): Nowadays that Nochrim do not serve idolatry, we are lenient about what they moved without intent (and all the more so for a Mechalel Shabbos). This is no source to permit what he poured. The custom is to be stringent even about what he poured, even if he did not touch the wine itself (Har Tzvi). Hotels must ensure that waiters who pour wine (or set the tables and open the wine) are Shomer Shabbos. One must be stringent if a Mechalel Shabbos shook an open bottle, but not if the bottle was sealed. One may ask him to bring a corked bottle through a public place, for he fears to open it lest people see him, and he would need to puncture or bend the cork and it would be evident that he opened it. Nowadays, one seal suffices even for Nochrim!

Yabi'a Omer (5 YD 10:3): A Mechalel Shabbos in public is like a Nochri regarding wine, but not regarding bread or Bishul Akum.

Shevet ha'Levi (2:51): The Mechaber holds that merely boiling wine makes it Mevushal. One should not rely on this to permit Nochrim to touch it, but it is enough for a Yisrael Mechalel Shabbos in public.