“Only God can judge me” is the typical response unbelievers and immature Christians give when confronted about their sin. Basically, they want you to validate them or leave (and in some cases, validation is the only option). But, as my mom once joked, “He left a note”. God didn’t leave us with no possible way to interpret His will. He didn't leave us to guess, He didn't leave us to do what we want and assume He's ok with it, He left us with a comprehensive collection of inspired writings, telling us everything we need to know to reprove, correct, and instruct each other (1 Timothy 3:16-17), and even to judge righteous judgement (John 7:24).
One thing to consider is that “only God can judge me” usually comes from sexually immoral people. 1 Corinthians 5 speaks specifically to this kind of person, and how we, as a Church, are to respond to them. Paul was made aware of a man within the church in Corinth who was sleeping with his father’s wife. Paul tells us that he has already judged this person, and his command to the congregation was to deliver that man to Satan for the destruction of his flesh, so that his spirit might be saved.
Paul continues, warning us not to associate with the sexually immoral, and lists other sinners we should not allow within the Church (but he does clarify that it’s different with unbelievers, since we all share the same planet, so we can’t really avoid them). How can we “put away from yourselves the evil person” (v13) if we don’t judge them as evil people?
If you believe in God, even the statement "only God can judge me" should be terrifying to you, for "It is a fearful thing to fall into the hands of the living God." (Hebrews 10:31). Look back at verse 5. “...deliver such a one to Satan for the destruction of the flesh, that his spirit may be saved in the day of the Lord Jesus.” What Paul is saying here is “look, this guy’s on a path to Hell. You need to get him off it.” Jesus describes this man’s potential fate in Matthew 7:21-23, where people call Him Lord, but He tells them “I never knew you; depart from me, you who practice lawlessness!” Now tell me, which would you prefer? The loving rebuke of a concerned brother, or the irreversible judgement of a Holy God?
In 1 Corinthians 5:12-13, Paul also tells us that God judges those who are outside the Church (i.e. unbelievers). Their fate is no better. In fact, it is the same: the lake of fire (Revelation 20:15). Everyone who loves unbelievers needs to preach the same message as the Apostles: all have sinned, all need to repent. See, God already has judged you. You’re guilty. You’ve broken so many of His laws that even if He could let just one of them slide, you’d still fail miserably. If you get what you deserve, you won’t be as quick to declare that God’s judgement is the only one you’ll accept. But the Bible gives us the answer: repent. Turn from your sin and place your life in the hands of your Creator. If you do that, there is no condemnation waiting for you (Romans 8:1). Fail that, you’re a dead man walking (John 3:18).