Does the person who wronged me, falsely accused me, and refused my reconciliation have any rights over me?

You have done well by initiating reconciliation, and the best of two disputing parties is the one who begins with greetings of peace. If you have not wronged him, then you owe him nothing. But if a shortcoming originated from you, then apologize to him, unless you fear a greater harm. In that case, seeking Allah's forgiveness and supplicating for him is sufficient. It is permissible to ward off suspicions from yourself, and to speak ill of the other only to the extent that he spoke ill of you, provided there is certainty and no exceeding of limits or slandering. The aim of Sharia (definition) is harmony, affection, goodly suspicion, and accepting excuses.