According to Islamqa, if a Muslim transgresses against the rights of others, then it is essential for him to ask the wronged person for forgiveness:
It says in al-Mawsoo’ah al-Fiqhiyyah: Repentance in the sense of regretting what one has done and resolving not to do it again is not sufficient to waive the rights that are owed to other people. Whoever has stolen another person’s wealth, or seized it by force, or has wronged him in any other way, will not have finished with the matter by simply regretting it, giving up the sin and resolving not to do it again. Rather he has to restore those rights. This is a principle on which the fuqaha (jurists) are unanimously agreed. End quote.
This has to do with material possessions, such as wealth that has been taken by force or by deceit. With regard to intangible rights such as in cases of slander and backbiting, if the person who has been wronged knows about it, then it is essential to apologize to him and ask for his forgiveness. If he does not know, then he should not be told; rather the one who wronged him should pray for him and pray for forgiveness for him, because telling him may upset him and create enmity and hatred between the two.
However, if the wronged person doesn't forgive the transgressor after the latter asks the former for forgiveness, what can the latter do for the expiation of that sin?