Praise be to Allah.

Firstly: 

Undoubtedly impugning a Muslim’s honour, making false accusations against him and lying are major sins and deeds that doom a person to Hell. Perhaps Allah will punish the one who does that with regard to his own honour, with things that will upset him, as requital for his evil and abhorrent actions. 

Abu Dawood (4876) narrated from Sa‘eed ibn Zayd, from the Prophet (blessings and peace of Allah be upon him) that he said: “What is more sinful than the worst type of riba is slandering a Muslim’s honour for no good reason.”

Classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

If reviling a Muslim is evildoing, then how about impugning his honour and saying things about his wife that are not true?! 

Secondly: 

If it is proven, with evidence, that he is accusing your chaste Muslim wife of immoral actions wrongfully, then he is a faasiq (evildoer) and the ruler should carry out on him the hadd punishment for slander, which is eighty lashes, and his testimony should be rejected unless he repents. Allah, may He be exalted, says (interpretation of the meaning):

“And those who accuse chaste women, and produce not four witnesses, flog them with eighty stripes, and reject their testimony forever, they indeed are the Fasiqoon (liars, rebellious, disobedient to Allah).

Except those who repent thereafter and do righteous deeds, (for such) verily, Allah is Oft-Forgiving, Most Merciful”

[an-Noor 24:4-5]. 

Shaykh as-Sa‘di (may Allah have mercy on him) said: 

Just as Allah, may He be exalted, regarded the case of the zaani as serious, and made it obligatory to flog him, and to stone him if he is previously married, and He ordained that it is not permissible to take him as a companion or mix with him in such a way that one may not be safe from his evil, He regarded accusing others of zina as serious too, as He said: “And those who accuse chaste women” i.e., free, chaste women – and the same ruling applies to men too, because there is no differentiation between the two. What is meant by accusing is accusation of zina, based on the context.

“and produce not four witnesses” i.e., men of good character, who can clearly testify to that.

“flog them with eighty stripes”. 

End quote from Tafseer as-Sa‘di

Thirdly: 

Accusing a chaste Muslim man of being a pimp requires a disciplinary punishment from the ruler or whoever is acting in his stead. An-Nawawi (may Allah have mercy on him) said: 

Among the verbal infractions that require a disciplinary punishment are when one person calls another an evildoer, a disbeliever, an immoral person, a doomed person, a dog, a donkey, a billy goat, a Raafidi, evil, a liar, a traitor, a pimp, or a cuckold.

End quote from al-Majmoo‘ (20/125) 

Al-Mirdaawi (may Allah have mercy on him) said: 

A person may be given a disciplinary punishment for calling another person a disbeliever, an immoral person, a donkey, a billy goat, a Raafidi… a pimp, and the like.

End quote from al-Insaaf (10/217) 

Fourthly: 

Deeds such as that which you describe are usually only done by foolish or promiscuous people, or evildoers who strive to spread mischief in the land. So do not pay any attention to what he says, and do not let yourself be distracted by it, because it is only the foolish words of a foolish person. All that you can do is to notify the authorities about him, so that they can track him down and take measures to deter him and others like him. 

With regard to what you have mentioned about killing him, if you can find him, by doing that you will be jumping out of the frying pan into the fire, and dealing with one wrong by committing a wrong that is greater than it. Cases of slander are not punishable by killing or execution; rather the hadd punishment that is prescribed in sharee‘ah is what we have mentioned above. Moreover, when has killing been a remedy for such matters? Undoubtedly you will make matters worse and cause greater harm. Do not stoke the fires of enmity in your heart, and fail to see the proper way of handling the matter. 

What we advise you to do is not to allow this annoyance to spoil your thinking and damage your affairs. Such things are done by foolish people. Cut off all means of communication with such people, even if that is by changing your phone number or not replying to strange numbers, and the like. 

You have the best example in your Prophet (blessings and peace of Allah be upon him). The hypocrites impugned the honour of his chaste wife ‘Aa’ishah bint Abi Bakr, whose innocence was declared from above seven heavens, and concerning whom the ummah is unanimously agreed that anyone who slanders her is a disbeliever. Yet despite that, the Prophet (blessings and peace of Allah be upon him) bore this great annoyance and hardship with patience. He did not kill anyone who indulged in that gossip; rather he carried out on them the hadd punishment prescribed by Allah, as Abu Dawood (4474) narrated from ‘Aa’ishah (may Allah be pleased with her) that she said: When my vindication was revealed, the Prophet (blessings and peace of Allah be upon him) stood on the minbar and mentioned that and recited Qur’an. When he came down from the minbar he ordered that the two men and the woman be flogged as a hadd punishment 

Classed as hasan by al-Albaani in Saheeh Abi Dawood. 

For more information, please see the answers to questions no. 112134, 954 and 108955 

And Allah knows best.