May one greet a gentile on the day of his holiday?
It is forbidden to enter the home of an idolater on the day of his Holiday and wish him Shalom/peace. If one found the gentile outside of his home, he may greet him with a low voice and melancholy demeanor. [Some Poskim rule the above prohibition only refers to using the word “Shalom” being that it is the name of Hashem, however, one may greet him using other terms. Practically, although one is to initially avoid greeting an idol worshiper on the day of their holiday, if one knows the gentile, or not doing so can cause enmity, then one may greet the gentile with a greeting other than Shalom, and even Shalom may be recited in a low voice if he meets him, as stated above.]
Q&A on greetings on Christian Holidays
*Christianity is considered idol worship according to Halacha, and thus the following questions relating to the above law are relevant:
May one wish a merry Chris-mas, or happy holidays, to a gentile acquaintance or neighbor?
Merry chris-mass: The term Chris-mas is not to be mentioned, due to the prohibition against mentioning the name of idols.
Other greetings: If the gentile does not believe in the religious connotations behind the holiday, then there is no prohibition to mention to him “Happy Holidays.” If, however, the gentile believes in the idolatry related content behind Christmas, seemingly, one may only do so in a pressing situation, to avoid enmity. Certainly one should not go out of one’s way to greet him and send him Holiday wishes, such as through social media, unless lack of doing so will cause enmity. In all cases that one meets a gentile acquaintance outside who is a practicing Christian, he is to greet him with a low voice, as stated above.
May one wish others a happy New Years on the 1st of January?
There is no prohibition involved in wishing a happy New Years to a gentile who does not affiliate the day with any Christian connotations or worship of a deity. One is to avoid wishing a happy New Years to a practicing Christian [particularly Catholics, and Lutherans] who believes in the Christian doctrine behind the New Year’s Holiday. However, even in such a case, one may do so in a pressing situation, in order to avoid causing enmity and anti-Semitism. Certainly one should not go out of one’s way to greet him and send him Holiday wishes, such as through social media, unless lack of doing so will cause enmity. In all cases that one meets a gentile acquaintance outside who is a practicing Christian and believes in the holiday of New Years, he is to greet him with a low voice, as stated above. [In general, it is not customary of Jews to wish other Jews a Happy new year on the first of January. However, it is related, that Rebbe Levi Yitzchak of Berditchiv would wish others a Happy new year on the 1st of January. Likewise, the Rebbe once wished a Chassid a happy new years on the morning of January first, in continuation of the tradition from Rebbe Levi Yitzchak. This is based on the verse in psalms 87:6 ““Hashem Yispor Bichsov Amim..”]
 Michaber 148:9; Gittin 62b
 The reason: As we suspect that this level of closeness of going to the home and greeting him can bring unwanted closeness between him and the gentile’s idolatry. [Bach 148:10]
 Bedek Habayis 148 in name of Orchos Chaim Ovada Zara 21 in name of Maharam, brought in Shach 148:7 and Nekudos Hakesef on Taz 148:6; [The Shach however leaves this matter in question]; See also Taz 148:6 in name of Semak 133 regarding the prohibition to repeat Shalom that it only applies to the word Shalom; Practically, see Birkeiy Yosef 148 in name of Maharikash regarding repeating Shalom that the custom is to repeat a blessing that does not have the name Shalom and seemingly the same would apply here that one may enter the home of a gentile and greet him with other words, or that he may greet him outside in a normal tone if he does not say the word Shalom. [To note however that although the Shach ibid records this ruling of the Bedek Habayis regarding this Halacha discussing greetings on the day of the Holiday, in truth it was said regarding repeating the word Shalom, and perhaps it is limited to that case, and not to the case under our discussion which prohibits entering the home on the day of his holiday and greeting him.] Vetzaruch Iyun.
 See Rama 149:12 and Terumos Hadeshen 195 regarding presents and the same would apply here
 Rambam Machalos Assuros 11:7; Avoda Zara 9:4; Pirush Hamishnayos Avoda Zara 1:3; Teshuvas Harambam 448; Rama Y.D. 148:12 [in uncensored editions] lists Xmas and New Years as Holidays of idolatry; Likkutei Sichos 37 p. 198; Rebbe in handwritten editing remarks to a letter “Christianity is Avoda Zara, is in contrast to the seven Nohadite laws, as opposed to Islam. However, the Christians of today are simply “Maaseh Avoseihem Beyadeihem”.
The reason: As they believe that Yoshka is one of the three parts of Hashem and they worship him. [In truth however, there are different sects of Christianity with different belief systems. See Haemuna Vehadeios of Rasag 2:7 that there are four groups of Christians and not all are idol worshipers; See here https://www.thoughtco.com/faith-groups-that-reject-trinity-doctrine-700367]
 See Teshuvos Rav Ezriel Hildsheimer 180 and Mishneh Halachos 9/169 regarding the word Christ that one is not to use this term as it connotes a Messiah and savior, and according to some even a deity, and so is the custom of all Jewry to not say this term. Seemingly, the same would apply to avoiding the word Christmas which is rooted in the word Christ and so is the custom. One is rather to use an epithet [i.e. nickname] such as Kratzmacht; Nittel, and the like. Seemingly however the term X-mass is not to be used, as the X is short for Ch***, and is used also by Christians as a formal name of the holiday. [See here http://www.thefreedictionary.com/Xmass]
 See Michaber 148:5 and Avoda Zara 65a regarding presents and the same would apply here; This is in addition to the Heter brought in Rama in next footnote
 See Rama 149:12 and Terumos Hadeshen 195 regarding presents and the same would apply here
 Historical background on New Years: The history of New Year’s celebrations date back to the Julian calendar [pre-Christendom], in which it was celebrated as a day of worship of the Greek deity called Janus, hence its name “January”, according to some historians. Later on, in the Gregorian calendar [Christendom], it was celebrated in Europe, and all Christian countries, as a Christian Holiday, commemorating the circumcision and naming of Yoshka. It was customary for the gentiles to receive gifts on this day as a good omen for the coming year. [Rama 149:12 [in non-censored editions]; Darkei Moshe Haaruch 148:5; Terumos Hadeshen 195] Until this very day, Catholic churches throughout the world hold a New Years mass, which is considered a day of obligation for Catholics. Protestants, however, do not necessarily view it as a day of obligation, do not hold mass, although many hold services on New Years. Practically, today, many gentiles do not affiliate New Year’s with any religious observance, and in fact are not even aware of the above history. It is simply a day to celebrate the start of the new year on the calendar, and make new year resolutions. Nonetheless, being that this holiday of New Years was affiliated with Christianity, which is defined as idolatry, the Poskim [Rama 149:12 [in non-censored editions]; Darkei Moshe Haaruch 148:5; Terumos Hadeshen 195; recorded in Likkutei Sichos 15/554; Shulchan Menachem 3/292] therefore discuss how Jews are to intermingle with gentiles on this day.
 Michaber 148:5 regarding presents, that they may be given to a gentile on his holiday if he does not worship idolatry; Avoda Zara 65a
 See Michaber 148:9 that one may not greet an idolater on the day of his Holiday unless he sees him outside, in which case he may greet him in a melancholy voice; See Shach 148:7 regarding if this applies only to the word Shalom, or other greetings; However, certainly saying the words “Happy Holiday” is more severe than simply saying good morning, as it gives credence to their idolatry, and hence should only be done for the sake of preventing enmity, as ruled regarding presents in Rama 149/12; Terumos Hadeshen 195
 Rama 149/12; Terumos Hadeshen 195 regarding presents
 The reason: As a) New Years is considered a Christian Holiday. And b) It denies the true New Year which is on Rosh Hashanah. Hence, we do not want to give credence to the Gentile new year. From the letter of the law however, seemingly there is no prohibition in doing so.
 Rabbi Sholom Hecht of Hecht’s bookstore on Coney Island in Flatbush, NY, merited to enter the Lubavitcher Rebbe’s inner chamber for “Yechidus.” One time Rabbi Hecht had a Yechidus with the Rebbe on the morning of January 1st. At some point during the Yechidus the Rebbe told him “Happy New Year”. Rabbi Hecht was very surprised. The Rebbe then told him that Rabbi Levi Yitzchok of Berdichev used to tell his congregants “Happy New Year”, and it’s based on Kapitel 87 verse 6 in Tehillim where it says “Hashem Yispor Bichsov Amim..” “Hashem will count in the register of people…” [Translation courtesy of Tehillim Ohel Yosef Yitzchok with English translation, by Kehot] The Rebbe is also recorded to have wished his secretary, Rabbi Nissan Mindel, a happy new Years on the 1st of January. So is also recorded of the Apter Rav, author of Ohev Yisrael, that he would wish a happy new year and bless the Jewish people on this day. The author of Baal Hayeshuos [Zlotchiv] would also bless the Jews with a good year, and would say with a smile that when Hashem sees how the gentile celebrate the new years, and compares it to the Jews, He tears their evil decrees.