REAL CHRISTIANS NEVER APOLOGIZE
Before I answer that, there is one thing we need to keep in mind: Scripture states that anytime we are aware of someone that has something against us, i.e. has a complaint against us, we are responsible to initiate the reconciliation. The Bible instructs us to pursue peace and to do all we can, “as much as it depends on you,” to live at peace with those we have offended. Jesus essentially says in Matthew 5:23-24, “Even if you are sitting in church and you realize someone has something against you, leave and do what you can to be reconciled with them.”
In the Bible, the primary means to seek reconciliation is through asking for forgiveness, not by apologizing. So what is the difference? Think about it: can you say, “Will you forgive me” with a nasty tone? It is almost impossible because forgiveness requires humility and contrition. Forgiveness is more complete because it requires you to say, “Will you forgive me for…” It requires you to admit and clarify the offense or the wrong committed. Forgiveness is the pathway to biblical reconciliation.
I can tell you, as my wife and I became more adept at removing apologies of any form from our vocabulary, and we started asking for forgiveness from each other and the spouse in turn forgiving in kind, our relationship struggles and periods of “intense fellowship” tended to become less volatile; the period of time our relationship was impaired didn’t last very long. I would encourage you to try it.
 Psalm 34:14; Romans 14:19; 2 Timothy 2:22; 1 Peter 3:11.
 Romans 12:18. "If possible, so far as it depends on you, live peaceably with all.”
 Matthew 5:23-24. “So if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there before the altar and go. First be reconciled to your brother, and then come and offer your gift.”
 Matthew 6:14-15; 18:21-33; Luke 6:37; Ephesians 4:32; Colossians 3:13.
 Psalm 51:17. “The sacrifices of God are a broken spirit; a broken and contrite heart, O God, you will not despise.”
by Michael Burner
An apology or asking someone for forgiveness is only acceptable to God when the offenders heart is remorseful and when the offender is willing to correct his error, if possible.
Wow, this is what you get from only looking at the Bible literally...immaturity! Saying sorry does a lot for reconciliation, but you've only used its examples in terms of it being said badly or insincerely. When said properly, with proper intentions it can be a wall smasher. It is also socially acceptable as the first part of reconciliation and forgiveness. It says many things, because there are many colours to the word sorry. It also shows maturity or ego. The less a person can say sorry ( and more excuses for doing that), the more egotistical that person is, or the less self esteem they have. The more appropriately a person can apologise, the more maturity and humility they show. What you're saying about forgiveness is good, but without sorry, its teaching to just follow a principle or command for its own sake and that doesn't allow for any sincerity, or the ability to be humble. Sorry may not be in the Bible, but neither are cars, but we still use them!