Is it sufficient to ask a friend for forgiveness without telling him the reason, knowing that he said: "I will ask God to exact my right from you on the Day of Judgment," and "May God forgive you"?

That he will ask Allah to take his right from you indicates that he has not forgiven you. If what you committed against him is a prevalent mischief by informing him of it, then suffice with seeking forgiveness and supplicating for him, and do not inform him. But if, were you to inform him, no mischief would occur, then apologize to him and ask for forgiveness. And if he forgives you without knowing the full extent of his right, we hope that this will benefit you.