Question: Can we greet non-Muslims by saying Assalamu ‘alaykum? 
    Answer: Greeting a non-Muslim is no different from greeting a Muslim. The Prophet (sws) himself said Assalamu ‘alaykum to the non-Muslims of his own times. Uthamah Ibn Zayd reports: 

The Prophet (sws) once passed by a mixed gathering of the Muslims, the Idolaters and the Jews and said Assalamu ‘alaykum to them. (Bukhari, Kitab al-Jihad)

    However, some people on the basis of the following Hadith are of the opinion that non-Muslims should not be greeted with Assalamu ‘alaykum: 

When the People of the Book greet you, reply them by saying wa alaykum. (Bukhari, Kitab al-Isti’dhan)

    If all the texts of this Hadith are collected and analyzed, what comes to light is the fact that the People of the Book used to mischievously twist their tongue in a swift manner and instead of saying the proper words of the salutation would say Assalamu ‘alaykum (Death be to you all), which when quickly pronounced sounded very near to the original words of Assalamu ‘alaykum. It was to counter this devilish prank that the Muslims were directed to say: Wa ‘alaykum (To you too). It must be borne in mind that the Muslims were asked to give this particular reply only after the truth had been revealed in its ultimate form by the Prophet (sws)to the People of the Book after many years of propagation, and in spite of that they were not willing to submit to it. Had this ultimate stage not come, one can safely conclude that Muslims would never have been asked to reply with equally harsh words and would have continued to say the words: Assalamu ‘alaykum, the best of prayers and the best of wishes. 
    Today no Muslim preacher can speak with the same authority as the Prophet (sws), and neither can he speak on behalf of the Almighty as the Prophet (sws). Therefore, he can never be in a position to say that the truth has been unveiled in its ultimate form. He must continue his efforts of propagation all his life, and remain a well-wisher of all the non-Muslims of the world^*, and continue to greet them with Assalamu ‘alaykum. 
  

^*. Consequently, the following Hadith (which contains a similar directive) also relates specifically to the People of the Book of the Prophet’s times on whom the truth had been unveiled by the Almighty through His last Prophet (sws). It has no bearing with the People of the Book of our times: 

Don’t initiate the of saying salutations to the Jews or the Christians, and force them to the narrowest of paths when one of them meets you on your way. (Muslim, Kitab al-Salam)