Understanding Matthew 7:1 in context
Matthew 7:1 is both the most well-known and the most misunderstood verse in the Bible, because when you read all of Matthew 7 there is clearly no prohibition on judgment. I, fact, exactly the opposite is true - Jesus commands us to judge in that chapter!
When Jesus tells us not to judge, he is telling us not to judge as hypocrites. After all, we cannot see clearly to remove a speck in someone's eye when we have a tree in our own eye. But Jesus does not command us not to remove the other person's speck - He commands us to remove the tree stuck in our own eye so we can see clearly to remove the speck in our brother's eye.
In fact, as you read through Matthew 7, there are clear commandments to judge that follow the warning against hypocrisy.
Verse 6 tells us not to cast pearls before swine or to give what is holy to the dogs. (It is important to note that Jesus called a Gentile woman a dog in Matthew 15:26-28.) But how are we to know who are the swine and the dogs? The only way to know that is to judge. Jesus follows up by warning against false prophets who seek to turn people away from the Gospel. How are we to know who these false prophets are? We know them by their fruits - meaning that we judge their fruit!
There are many other places in Scripture where we are commanded to judge, including the Apostle Paul's rebuke of the church at Corinth for allowing a man to commit adultery with his father's wife. Paul commands them to remove the man committing incest from the church. I could go on and on with many more examples that further shatter the illusion we have created around Matthew 7:1.
The reason we intentionally misunderstand Matthew 7:1 is that we have rejected the authority of the church and that we want to continue in our sin without being held accountable. But we do not even truly believe the nonsense we repeat about judging - after all, if someone wrongs us we want our pound of flesh. No one would suggest that murderers and rapists be allowed to get away with their crimes because of a perverted misinterpretation of Matthew 7:1.
It is not enough to cherry-pick Scripture for things that we can pervert to justify and cover our own wickedness. We need to love His word, His church and His people, and strive to obey him out of that love.