Of course we should apologize when we are wrong
Apologizing when you are wrong is not a sign of "weakness." It is a sign of humility, strength and faith.
In recent years, many conservatives (including far too many Christians) have adopted a "never back down, never apologize" mindset. This mentality is something that has existed for some time, but it has become much more common over the last decade. While there are legitimate principles behind it, "never apologize" is a wicked mindset that is profoundly anti-Gospel and anti-Christian. God warns us in His Word:
Proverbs 29:1
He, that being often reproved hardeneth his neck, shall suddenly be destroyed, and that without remedy.
Do I really need to say this? Obviously, this does not mean that you should apologize when people are offended by Christian doctrine or because you said things that are true. This his does not mean that you should allow yourself to be victimized or allow someone to hurt your family or loved ones. That is a childish response. It is our habit to always look for the exception to a rule or moral standard, and then claim that the entire standard is wrong. This should not be.
There are plenty of times where conservatives say and do things that are wrong. For Christians, it is absolutely essential that we recognize when we are wrong. This extends from violations of basic manners to much more serious matters. If you break God's law, that is the time to back down and apologize. The most egregious examples I have seen is when a Christian (and sometimes a "christian") makes a false accusation and then refuses to correct the record when proven wrong. Making an innocent mistake may be understandable, but refusing to correct it is wicked. Remember that God says that He hates lies twice in Proverbs 6:16-19 and that warning is repeated in Proverbs 12:22. God provides a chilling warning for liars in Revelation 21:8.
Yes, I understand that far too many people have "apologized" to the woke mob after saying something truthful. Apologizing for telling the truth is sinful. You should not do that, obviously. We should obviously never back down or apologize for speaking the Gospel message of forgiveness and redemption for repentant sinners, but that is not the same as apologizing and repenting for breaking God's law.
We are all wrong at times. Every one of us. When we are wrong, we should repent and apologize, rather than refusing to admit fault out of some sort of deranged fear of being "weak." As Scripture warns, persisting in this stubborn pride will lead to your destruction. It is not "strong" or "courageous" to double and triple down when you are wrong. It shows weakness of character. It shows you put your pride over the truth and even over God Himself. Repent.