And have you forgotten the exhortation that addresses you as sons?
“My son, do not regard lightly the discipline of the Lord,
nor be weary when reproved by Him.
For the Lord disciplines the one He loves,
and chastises every son whom He receives.”
It is for discipline that you have to endure. God is treating you as sons. For what son is there whom his father does not discipline? If you are left without discipline, in which all have participated, then you are illegitimate children and not sons. Besides this, we have had earthly fathers who disciplined us and we respected them. Shall we not much more be subject to the Father of spirits and live? For they disciplined us for a short time as it seemed best to them, but He disciplines us for our good, that we may share His holiness. For the moment all discipline seems painful rather than pleasant, but later it yields the peaceful fruit of righteousness to those who have been trained by it. (Hebrews 12: 5-11)
The writer to Hebrews is clear that God's discipline is an act of love.
Writing to Titus, the Apostle Paul informs him:
As for a person who stirs up division, after warning him once and then twice, have nothing more to do with him, knowing that such a person is warped and sinful; he is self-condemned.(Titus 3: 10-11)
With regards to the Corinthian church, Paul gives strong instructions:
It is actually reported that there is sexual immorality among you, and of a kind that is not tolerated even among pagans, for a man has his father's wife. And you are arrogant! Ought you not rather to mourn? Let him who has done this be removed from among you.
For though absent in body, I am present in spirit; and as if present, I have already pronounced judgment on the one who did such a thing. When you are assembled in the name of the Lord Jesus and my spirit is present, with the power of our Lord Jesus, you are to deliver this man to Satan for the destruction of the flesh, so that his spirit may be saved in the day of the Lord.
Your boasting is not good. Do you not know that a little leaven leavens the whole lump? Cleanse out the old leaven that you may be a new lump, as you really are unleavened. For Christ, our Passover lamb, has been sacrificed. Let us therefore celebrate the festival, not with the old leaven, the leaven of malice and evil, but with the unleavened bread of sincerity and truth.
I wrote to you in my letter not to associate with sexually immoral people — not at all meaning the sexually immoral of this world, or the greedy and swindlers, or idolaters, since then you would need to go out of the world. But now I am writing to you not to associate with anyone who bears the name of brother if he is guilty of sexual immorality or greed, or is an idolater, reviler, drunkard, or swindler — not even to eat with such a one. For what have I to do with judging outsiders? Is it not those inside the church whom you are to judge? God judges those outside. “Purge the evil person from among you.” (I Corinthians 5)
There is a place for church discipline. This is quite strong, and not language you'd hear in church these days - "deliver this man to Satan".
Basically, put him outside the church and the spiritual protection it provides. Let him follow the life he wants and see where it leads. Such exclusion is reserved for serious matters which will damage the church. Things not given in the New Testament as grounds for exclusion include not being cool, not being financially successful, being working-class, not being sporty, being in poor health etc.
But the aim is shown in Paul's follow-up letter to Corinth:
Now if anyone has caused pain, he has caused it not to me, but in some measure — not to put it too severely — to all of you. For such a one, this punishment by the majority is enough, so you should rather turn to forgive and comfort him, or he may be overwhelmed by excessive sorrow. So I beg you to reaffirm your love for him. For this is why I wrote, that I might test you and know whether you are obedient in everything. Anyone whom you forgive, I also forgive. Indeed, what I have forgiven, if I have forgiven anything, has been for your sake in the presence of Christ, so that we would not be outwitted by Satan; for we are not ignorant of his designs. (II Corinthians 2: 5-11)
Church discipline should have the endpoint of repentance, restoration and reconciliation.
There are two ways church discipline is done. One is the formal, Biblical model, which meets the standards of justice, and provides excommunication (e.g. the Book of Common Prayer's rubrics do outline when someone should not be allowed to share in the Lord's Table) with the aim of reconciliation. The other is church discipline by a spiritual kangaroo court, with its whispers and gossip, with the target not being able to respond, and its excommunication by a thousand cuts, with the aim that the person gossipped about leaves and never comes back.
If a church feels that the Biblical method of church discipline is somehow unworthy and unloving, then its discipline will be exercised by the other method. And the people who will end up being disciplined won't be the ones that the New Testament says should be disciplined.