Amazing Idea Kween! Well, here goes my opinion;
I think that most times ego is the main culprit when it comes to not apologizing. In as much as saying sorry is practically the first thing we learnt as kids while growing and developing our characters, we may have somehow picked characters which aren't necessarily the best and have somehow interfere with the first values we learnt. Egoistic people find it hard to say words like "I love you or I'm sorry", which is quite sad if you ask me.
There are people, however, who don't apologize because they believe apologizing will make them look vulnerable and too emotional than they want to appear. For others, they fear that when they accept their wrongs and apologize, it paves way for more accusation or blames from the person they are apologizing to. The reasons really are much! I believe it might just be a way of "protecting" themselves from what doesn't need protection. What apologies do is strengthen and mend relationships, who need protection from what will make relationships between friends, families, couples etc. blossom?
I have learnt to tell myself "sorry" even before people that wrong me apologize and it has gone a long way in helping me to care less about people apologizing to me or not. Sometimes friends might hurt us without meaning to or maybe they don't understand how important it is to apologize, I try my best to let them know but when my effort seems futile, I just forget it. We can't force them to change, but we can at least help them grow.

Well, I think apologies are an extremely important part of any relationship. It opens an opportunity for forgiveness which in turn brings about reconciliation and a peaceful coexistence. It also makes it easier for the person that had been hurt to deal with issues. A world where no one apologizes will be one full of chaos and sadness, which isn't what our world should be like.

It not all about who is right or who is wrong, it is about staying peacefully and making others happy as well, apologizing won't costs us much, but it gives us back a whole lot of benefits. (hope its not too long, I dunno how it ended up this long. lol.)