Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
To break off relations with a Muslim is a sin but not an act that drives the person out of Islam. It is stated in the Alsaheehayn as well as in other books that to break off relations with a Muslim is prohibited. Anas bin Malik told that the Prophet (Sallallahu Alaihi wa Sallam) said: "(O! Muslims) Do not hate each other, do not envy one another, and do not be hostile to one another (standing back to back), rather be brothers (as) servants of Allah. A Muslim is not allowed to desert his brother (Muslim) more than three days." This means, as explained in the Sunan: To avoid whatsoever may cause hatred, and none of you should wish the termination of other's amenity, whether you wish it for yourself or not, and do not break off from one another more than three nights with their days.
Since human beings get angry by nature, desertion is accepted for only this (limited) time and less (than that) in the hope that he will come back to his senses and will return. This regards relations between Muslims and shortcomings that could affect these relations such as not dealing kindly with one another. But if the cause for severing ties is religious, then one should not regret it. (On the contrary), a Muslim is required to break off from the people of innovation and of those who do things as they like, unless they repent and come back to their consciousness and get themselves right. But to mix with him is better if it is hoped that this will result in their accepting to be guided to the right path.
It has also been stated in Sunan Abu Dawood by Abu Hurairah that the Prophet (Sallallahu Alaihi wa Sallam) said: "The Muslim is not allowed to abandon a Muslim brother more than three (nights). If three days pass and he meets him, he should greet him. If he (the other believer) returns his greeting, then they share the reward; but if he does not return it, he (alone) will incur the sin." Ahmad added: 'And the Muslim is then freed from the desertion.' In another Hadith (it says): "He who breaks off from his (Muslim) brother more than three and dies (during this period) will go to Hell." In another Hadith (it says): "He who deserts his (Muslim) brother for a year is like shedding his blood."
Also, Abu Hurairah reported that the Prophet (Sallallahu Alaihi wa Sallam) said: "The doors of Heaven are opened every Monday and Thursday, and Allah pardons, in these days every individual servant who is not a polytheist, except those whose enmity is between them; Allah Says: 'Postpone those until they reestablish normal relationship (with each other)."
Therefore, your brother must turn to Allah and stop breaking off from you, and you should not cut contacts with him: Greet him whenever you meet him, visit him if ever he gets sick, and give all the rights that Allah made obligatory on a Muslim for his brother Muslim. Let him get away with the sin alone if he insists on not replying to greetings.
May Allah lead us and you to do the best things that earn His pardon and forgiveness, as He is able to do all things. Ameen.
Allah knows best.