Short answer
Yes. A non-Muslim may say as-salamu alaykum (peace be upon you) to a Muslim, and most Muslims will be touched by it. The usual reply is wa alaykum as-salam (and upon you be peace). A simple hello and a smile are just as good.
Nothing forbids a non-Muslim from using the greeting, and the Muslim should reply, since the Quran commands returning a greeting. The Hanafi texts say to reply to a non-Muslim with wa alaykum (and upon you); some contemporary scholars allow the fuller reply when the greeting is clearly sincere.
The four schools agree on this.
Explanation
As-salamu alaykum means peace be upon you. It is a short prayer for safety and blessing, and Muslims use it in every language and country. Saying it to a Muslim friend, colleague or neighbour is a kind gesture, not an intrusion into their religion.
The Prophet ﷺ taught Muslims to greet people they know and people they do not know, and the Quran tells them to answer any greeting with one better or at least the same. So a Muslim will answer you.
Do not be surprised if a Muslim answers with just wa alaykum (and upon you). That short reply comes from a hadith about greetings from Jews and Christians in Madinah, and some Muslims follow it closely. It is not meant as coldness. Many will give the full reply, wa alaykum as-salam.
Other good manners when greeting Muslims:
- Handshakes: many Muslims do not shake hands with the opposite gender. Let them offer first; a smile or a hand on the heart is fine.
- Festivals: Eid Mubarak (blessed Eid) and Ramadan Mubarak are warmly welcomed.
- Hello is fine: no Muslim expects a non-Muslim to use Arabic.
For the Muslim side of the question, see can you say salam to non-Muslims, and for other phrases see Islamic phrases and their replies.
The evidence
Quran, Surah An-Nisaa 4:86
وَإِذَا حُيِّيتُم بِتَحِيَّةٍۢ فَحَيُّوا۟ بِأَحْسَنَ مِنْهَآ أَوْ رُدُّوهَآ ۗ إِنَّ ٱللَّهَ كَانَ عَلَىٰ كُلِّ شَىْءٍ حَسِيبًا ﴿٨٦﴾
And when you are greeted with a greeting, greet [in return] with one better than it or [at least] return it [in a like manner]. Indeed, Allah is ever, over all things, an Accountant.
A Muslim must return a greeting, whoever gives it.
Sahih al-Bukhari 12
Sahih... تطعم الطعام وتقرأ السلام على من عرفت ومن لم تعرف ...
"Feed people, and greet those you know and those you do not know."
Spreading greetings of peace is among the best parts of Islam.
Sahih al-Bukhari 6258
Sahih... إذا سلم عليكم أهل الكتاب فقولوا وعليكم ...
"When the People of the Book greet you, say: and upon you."
The source of the short reply some Muslims give.
Where the ruling is recorded
- al-Fatawa al-Hindiyya, Kitab al-Karahiya, the chapter on greeting. Discusses replying to the greeting of a non-Muslim.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.