When you read Galatians you can tell that Paul was very worked up when he wrote it. Was he just an emotional guy who flew off the handle with little provocation or was the situation genuinely serious? One way we can investigate this question is to compare it to other letters that Paul wrote where he clearly addressed problems in the church or got agitated.
6 I am astonished that you are so quickly deserting the one who called you by the grace of Christ and are turning to a different gospel— 7 which is really no gospel at all. Evidently some people are throwing you into confusion and are trying to pervert the gospel of Christ. 8 But even if we or an angel from heaven should preach a gospel other than the one we preached to you, let that person be under God's curse! 9 As we have already said, so now I say again: If anybody is preaching to you a gospel other than what you accepted, let that person be under God's curse!
10 Am I now trying to win human approval, or God's approval? Or am I trying to please people? If I were still trying to please people, I would not be a servant of Christ.
One instance to consider is Romans 14 and 15. Here there was a dispute within the church over Sabbath keeping and dietary restrictions. Even though Paul takes sides in this dispute (with the ‘strong’), his tone throughout is cordial and ultimately rules in favor of both parties. Both can practice as they wish as long as they don’t judge or stumble the other side. Paul puts unity ahead of being right.
We have another case is in 1 Corinthians. Here Paul deals with a litany of issues ranging from factionalism to the condoning of an incestuous relationship. Paul certainly expresses shock and is sarcastic at times, but again he never throttles them the way he does the Galatians. At times he rebukes them, but it’s clear, again, that he’s primarily seeking to restore unity in the church.
Philippians comes to mind when one looks for examples of strong language from Paul. In chapter 3 he calls a group that sounds a lot like the Teachers in Galatia, ‘dogs.’ Paul also seems to be somewhat agitated in 2 Corinthians, but this seems to be primarily based on personal attacks and then he never uses the vitriolic language that we find in Galatians.
Clearly one thing that is very important to Paul is unity. On non-essential issues, being right is secondary to promoting unity. Where does unity lie? It lies in the gospel. The gospel is the dividing line for Paul. With fellow believers we must have unity, but we must not be united with outsiders, ‘Do not be yoked together with unbelievers. For what do righteousness and wickedness have in common?’ (2 Cor. 5:14 - TNIV). That’s why he can be so harsh in Galatia. He’s seeking to preserve the unity and purity of the church by driving a wedge between the Galatians and the false teachers because they are leading the Galatians away from God. He's trying to drive a wedge between his converts and the teachers, much like Jude did, and, as I mentioned in this post, Jude provides a softer approach to the wavering that we must not ignore. The main thing to remember, though, is that we too need to be vigilant to preserve the purity of the gospel, but beyond that we must promote the unity of the body and not divide or pick fights over smaller issues. We can still address those who are in error (in our opinion) in non-essential matters, Paul certainly does that plenty of times, but we must do it in a way that does not disrupt the unity of the church. Ideally, correction should serve unity.