People who have integrity are usually really humble and always trying to become better human beings. If they have done anything wrong or have hurt someone even unintentionally, they apologize honestly, and they take steps to correct their mistakes.
People who have integrity are always ready to admit that they did wrong and their apologies are sincere.
A sincere apology requires three things: that you accept responsibility, that you acknowledge the feelings of the other person, and that you explain what you will do to make sure something like that never happens again.
The notion that simply saying “I apologize” should wipe out responsibility without the need to change one’s behavior is abhorrent.