Never apologize to the outrage mob?
I see two conflicting things on apologies, both of which are flawed, if not outright wrong. The primary reason they are wrong is that they reject the hard work of discernment. They are intellectually lazy. One school of thought is that you should apologize if you hurt someone's feelings. The other is that you should never apologize because it shows weakness and encourages the mob.
To the first: It really depends on what you said that offended someone. Well, is what you said actually, objectively offensive? Did you speak an uncomfortable truth that made people angry? If it is the former, then an apology may be warranted. If it is the latter, you should not apologize. The idea that only the offended person can judge whether something is offensive is absurd. We should not be afraid to speak truth because some people do not want to hear it.
To the second: Again, is what you said actually offensive? Let's say you falsely accuse someone of committing an immoral or illegal act. Obviously, you should apologize. Perhaps what you said was demeaning to a group of people – not because they feel demeaned, but because you told a joke that was simply mean-spirited. Even if there was no malicious intent, an apology may be warranted. But apologizing for speaking truth, again, only feeds and encourages the outrage mob.
Basically, you have to use discernment. You need to honestly examine what you said or did, and whether it actually requires an apology. The "never apologize to the outrage mob" and "always apologize for hurt feelings" camps both refuse to use discernment, and both extremes abandon moral courage – either the moral courage to stand by truth or the moral courage to admit when one is wrong. You may need to seek counsel from someone you know and trust, because it is very easy to have your judgment be clouded by controversy.