Is the husband's statement to his wife's brother: "If your sister doesn't go home today, she will no longer be my responsibility," considered a divorce, knowing that the wife stayed at her father's house for 20 days and there was no dispute between them?

Your statement "you will no longer be under my responsibility" is a metaphor for divorce. If you did not intend it as a divorce, or if you intended it as a promise of future divorce, then no divorce has occurred. However, if you intended it to enact a divorce immediately upon your wife's failure to come home that day, then a divorce has occurred, and you may take her back (perform raj'ah) provided this is not the third divorce and her waiting period (iddah) has not ended. If her iddah has ended, she has become irrevocably separated from you, and she is not permissible for you except with a new marriage contract. If this divorce completes the three divorces, then she is not permissible for you until she marries another man, he consummates the marriage with her, and then he divorces her.