Is it truly forgiveness in its complete sense if the other does not want to reconcile?
True forgiveness (what this moral virtue is at its essence) is different from complete forgiveness (what the end point or goal of forgiveness is). Because forgiving is a moral virtue it is good in and of itself whether or not the other person responds favorably to your forgiving. Complete forgiveness occurs when the end point is reached. A major end point of forgiving is for the other to receive your gesture of good will, to have remorse, repentance, and if possible, recompense for the injustice. This, then, is followed by reconciliation. So, forgiving can take place in one individual as an unconditional outpouring of kindness, respect, generosity, and love toward the other. Complete forgiveness occurs when there is forgiving (on the part of the offended), receiving the forgiving (on the part of the one offending), and then reconciliation between the two.
For additional information, see Forgiveness Defined.