How to kick someone out of church
Finally then, brothers and sisters, we ask you and urge you in the Lord Jesus, that as you received instruction from us about how you must live and please God (as you are in fact living) that you do so more and more. For you know what commands we gave you through the Lord Jesus. For this is God’s will: that you become holy, that you keep away from sexual immorality, that each of you know how to possess his own body in holiness and honor, not in lustful passion like the Gentiles who do not know God. In this matter no one should violate the rights of his brother or take advantage of him, because the Lord is the avenger in all these cases, as we also told you earlier and warned you solemnly. For God did not call us to impurity but in holiness. Consequently the one who rejects this is not rejecting human authority but God, who gives his Holy Spirit to you.
Now on the topic of brotherly love you have no need for anyone to write you, for you yourselves are taught by God to love one another. And indeed you are practicing it toward all the brothers and sisters in all of Macedonia. But we urge you, brothers and sisters, to do so more and more, to aspire to lead a quiet life, to attend to your own business, and to work with your hands, as we commanded you. In this way you will live a decent life before outsiders and not be in need.
Paul covers the big one, sex, but couches it in love and tells us to mind our own business. Hence, I think it's not my job to find out if a discreet sinner is actually doing one of the big ones. Paul's complaint about Corinth was started by a guy living with his mother-in-law and everyone knew it and were proud of their tolerance.
So this is where Matthew 18 may play into this. Jesus wants us to keep it discreet. Two friends who presumably have each other's best interests in mind should help each other stay on the straight and narrow. They need to keep each other pointed heavenward. They also need to have enough love for each other that it will cover a multitude of sins.
What does it mean to cover sins? I thought about drunk and passed out naked Noah, who I wrote about before. One son mocked his father's sin and its consequences, but the other two covered him up.
There are abundant reasons to expose a sin and disfellowship a sinner immediately, usually involving trauma against someone else. This is where the church messes up a great deal. We are so discrete we never solve anything or we are never discrete and only 5 people at a time are members in good standing. I think 1 Thess. 4 is a great example of balance.