In Judaism, we have a specific viewpoint about mistakes. G-d did not create us perfect. If He had (or if She had, to be politically correct), life would be real boring, wouldn’t it? Nothing to do but sit around being perfect, which, by the way, many people would prefer! So in the Torah, there are all sorts of mechanisms for redemption.
According to Torah, if someone erred or unknowingly or unwittingly made an error, they can bring a sacrifice, go through the appropriate process, and they’re forgiven. Today, since we don’t have a Temple, we say the Amidah, the standing prayer at the center of each prayer service. Contained within the weekly Amidah is the prayer that states, “forgive me for I have sinned, pardon me for I have erred”, as we lightly beat our chest. And of course, each year, we observe Yom Kippur.
As an interesting side note, one of my mentors told me that when you ask G-d for forgiveness and go through the repenting process, G-d wants to see if you’re sincere. G-d will then put you in the same situation, or a similar one, to see how you handle it. In other words, of course, did you really learn something and grow, or is it just lip service?
In Judaism, apologies are taken seriously. Someone apologizes three times and that shows they’re contrite. If the person to whom they’re apologizing doesn’t accept on the third attempt, it’s now on them.
Consider, as well, the consequences of not accepting the apology and continuing to vilify the person even after they’ve attempted to make amends. The person is going to feel resentful, probably. They’re going to resent that they were vulnerable, and people took advantage to insult them even more. That person would probably think twice the next time a mistake is made (none of us are perfect and we all continue to do stupid things throughout our lives).
Whereas that person could have made an effort to change and be more diligent about how they treat others, now they see the world as more hostile. They will probably react as such and now we have another antagonist out there.
So let’s all be more understanding. G-d is both merciful and judgmental. Hashem accepts our contrition. Let’s follow His example.