No wonder that the greeting of Islam is peace, which is the greeting of people of Paradise. It is permissible for a Muslim to greet anyone with the greeting of Islam regardless of his religion, race, or ethnicity. Also, if a non-Muslim greeted a Muslim using the greeting of Islam (as-salamu `alaykum), then a Muslim is required to return his or her greeting.

Sheikh FaysalMawlawi, deputy chairman of the European Council for Fatwa and Research, stated:

1. It is possible for a non-Muslim to use the greetingas-salamu`alaykumwarahmatu-llahiwabarakatuh, but he or she would not receive a reward for it as a Muslim would. This is because this greeting is part of Prophet Muhammad’s Sunnah in which non-Muslims do not believe.

2. It is permissible for a Muslim to greet a non-Muslim using the wordsalam (Arabic for: peace). This view has been attributed to Ibn `Abbas, IbnMas`ud, IbnMuhayriz, `UmarIbn `Abd Al-`Aziz, SufyanIbn `Uyaynah, Ash-Shu`abi, Al-Awza`i, and At-Tabari. Moreover, this opinion was chosen by Rashid Rida in his TafsirAl-Manar and Ash-Shanqiti in his Adwaa‘ Al-Bayan.

We also support this view, on the basis of the following Qur’anic verses that adopt and recommendsalam (peace) without restriction.

O you who believe, enter not houses other than your own without first announcing your presence and invoking peace upon the folk thereof.] (An-Nur 24:27)

[And when they hear vanity, they withdraw from it and say, “Unto us our deeds and unto you yours. Peace be unto you! We seek not the ignorant.”] (Al-Qasas 28:55)

When the foolish address them, they answer, “Peace.”] (Al-Furqan 25:63)

And he [Prophet Muhammad] says, “O my Lord! Lo! Those are a folk who believe not.” Then bear with them (O Muhammad) and say, “Peace.” But they will come to know.] (Az-Zukhruf 43:88-89)

He [Prophet Ibrahim] said, “Peace be
unto you! I shall ask forgiveness of my Lord for you.” Lo! He was ever gracious unto me.
] (Maryam 19:47)

Furthermore, there are a number of authentic hadiths that also recommend peace without restriction; that is to say, they recommend it toward all people. The Prophet (peace and blessings be upon him) said,

“Greet those
whom you know and those whom you do not know.”
(Al-Bukhari and Muslim)

“When Allah
created Adam, He said (to Adam), ‘Go and greet that group of angels sitting there, and listen to what they will say in reply to you, for that will be your greeting and the greeting of your offspring.’ Adam (went and) said, ‘As-salamu `alaykum (peace be upon you).’ They replied, ‘As-salamu `alaykawarahmatullah (peace and Allah’s mercy be on you).'” (Al-Bukhari and Muslim)

“Spread peace.”
(At-Tirmidhi, who said it is hasansahih — Arabic for:agreeable and authentic — and by IbnHibban in his Sahih )

There is another authentic hadith on which the majority of scholars depend in supporting a different viewpoint: That it is legally detestable (in Arabic:makruh) or even prohibited (in Arabic:haram) for a Muslim to greet a non-Muslim with “As-salamu `alaykum.” In this hadith, the Prophet (peace and blessings be upon him) said, “Do not initiate (greeting) Jews and Christians with salam (Muslim).
this hadith was connected to an existing state of war at the time. This can be supported by other versions of the hadith, which are also authentic, such as the version according to which the Prophet (peace and blessings be upon him) said, “We are going early (in the morning) to (fight) Jews, so do not start (greeting) them” (Narrated by Ahmad and At-Tabarani with a trustworthy chain of narrators). That was on the occasion when the Prophet (peace and blessings be upon him) invaded BanuQurayzah (who were Jews).
According to another version narrated by Ahmad, the Prophet (peace and blessings be upon him) said, “Verily, I shall leave tomorrow to (fight) Jews, sodo not start (greeting) them.” This version is authentic, as mentioned in Al-FathAr-Rabbani.
Moreover, in FathAl-Bari, Al-`Asqalani quoted a narration of Al-Bukhari (recorded in Al-Bukhari’sAl-Adab Al-Mufrad) and which was also narrated by An-Nasa’i. It reads: The Prophet (peace and blessings be upon him) said, “Verily, I shall leave tomorrow to (fight) the Jews, sodo not start (greeting) them with salam.” The difference between the two versions is “the” before “Jews” in the second one.

This opinion, which is adopted by a large number of scholars and which says that it is permissible to greet a non-Muslim using the word “salam” (or its translation), is confirmed for those who live outside Islamic lands but in countries which are in a state of peace (with Muslims).
It can be regarded as a form of covenant with the people with whom a Muslim lives whether he or she is a citizen or only a resident. This is because the Muslim, in this case, is supposed to practice da`wah with them (as every Muslim should do), and initiating a greeting is one of the morals of preachers.

3. If, however, a non-Muslim greets a Muslim with the word “salam” [or its translation], it becomes necessary to answer him, because Almighty Allah says, [When ye are greeted with a greeting, greet ye with a better than it or return it…] (An-Nisa’ 4: 86) It is “recommended” (in this verse) to answer the greeting with a “better” one, yet it is “obligatory” to “return” it. Returning a greeting requires an equal greeting, as maintained by Ibn Al-Qayyim in “AhkamAhlAdh-Dhimmah”.