Abu Dawud

Hadith Commentary:
(1)
It was the filthiness and evil nature of the Jewish people that when they would greet the Messenger of Allah (sallallahu alayhi wa sallam) or any Muslim, they would vent their malice by greeting in such words:
"Death or destruction be upon you."
The Messenger of Allah (sallallahu alayhi wa sallam) would take note of this by returning their words upon them, and he (sallallahu alayhi wa sallam) also instructed his noble Companions (radi Allahu anhum) that instead of responding to their abuse with foul language, their own words should simply be returned to them.
In this manner, their curse would itself become a cause of misfortune and punishment for them.
If they openly commit insolence against the Messenger of Allah (sallallahu alayhi wa sallam), then they should be killed, just as the Messenger of Allah (sallallahu alayhi wa sallam) himself ordered the killing of Ka'b bin Ashraf and Abu Rafi', the Jews.

(2)
In the present era, those who insult and demean the Messenger of Allah (sallallahu alayhi wa sallam) by making caricatures of him, the same ruling applies to them, and this is an unforgivable crime, because there is no command from the Messenger of Allah (sallallahu alayhi wa sallam) himself to pardon them, as he is no longer present.
We have written a detailed fatwa on this subject, which can be found in the second volume of Fatawa Ashab al-Hadith.
And Allah is the One whose help is sought.

Hadith Commentary:
The background of this hadith is that the noble Companions (radi Allahu anhum) asked the Messenger of Allah (sallallahu alayhi wa sallam): "The People of the Book greet us with salam, so how should we reply to them?" He said:
"You should say to them, 'Wa alaikum' (And upon you)."
That is, do not use the word 'as-salam' (peace).
(Sunan Abi Dawud, Al-Adab, Hadith: 5207)
Rather, in a hadith narrated from Anas (radi Allahu anhu), there is further clarification that a Jew passed by the Messenger of Allah (sallallahu alayhi wa sallam) and said, "As-sam alaik," meaning, "Death be upon you."
The Messenger of Allah (sallallahu alayhi wa sallam) replied only with "Wa alaik" (And upon you), then he (sallallahu alayhi wa sallam) said to the noble Companions (radi Allahu anhum):
"Do you know what he said? He said, 'As-sam alaik.'"
The Companions (radi Allahu anhum) submitted:
"Should we not kill him?" He (sallallahu alayhi wa sallam) said:
"No, rather, when the People of the Book greet you in this manner, you should reply with 'Wa alaikum.'"
(Sahih al-Bukhari, Istitabah al-Murtaddin, Hadith: 6926)
Similarly, in another narration, when the Jew was apprehended and questioned, he admitted to this, so the Prophet (sallallahu alayhi wa sallam) said:
"Return to him what he said."
(Al-Adab al-Mufrad, Hadith: 1105)

Takhrij al-Hadith: [وأخرجه البخاري 6257، من حديث مالك، ومسلم 2164، من حديث عبدالله بن دينار به]
Jurisprudential Points
➊ One should not say "as-salamu alaykum" to disbelievers (kuffar). If they offer salam, then they should be answered with "wa alaykum."
➋ The Jews, Christians, and all disbelievers are staunch enemies of the Muslims, and in this enmity, they are all united.
➌ Responding to salam is obligatory and necessary.
➍ If any disbeliever commits explicit disrespect (blasphemy) against the noble status of the Prophet Muhammad (sallallahu alayhi wa sallam), then according to other evidences, he should be killed. See Shaykh al-Islam Ibn Taymiyyah rahimahullah’s famous book: "As-Sarim al-Maslul ‘ala Shatim ar-Rasul" (sallallahu alayhi wa sallam).
➎ A Yemeni man greeted Sayyiduna Ibn Abbas (radi Allahu anhu) after he had become blind, saying: "As-salamu alaykum wa rahmatullahi wa barakatuh," and after that, the man added some words. Abdullah ibn Abbas (radi Allahu anhu) said: "Salam ends at 'barakatuh.'" [الموطأ 2/959 ح1855، وسنده صحيح]
➏ If someone uses harsh language against another, then, keeping the limits of the Shariah in view, it is permissible to respond to him with severity.

Benefit:
From this, it is understood that if a non-Muslim offers greetings (says salam), then in response, one should only say "Wa 'alayk" (and upon you), and one should not initiate greetings (salam) to a non-Muslim.