Does a husband's statement to his wife, "If you go to your family's house, I will bring someone else to care for her," constitute a divorce, after an argument about caring for his mother, knowing that she retracted her statement and affirmed her care for his mother?

Your statement to your wife: "Go to your family's house" is a كناية (allusion/indirect expression) of divorce. If you intended divorce by it, then it is effective and you cannot retract it. If you did not intend it, then it is not a divorce. If divorce occurs, you may take your wife back before her 'iddah (waiting period) ends, provided this divorce does not complete the three divorces. The 'iddah of a divorced woman ends with her purification from the third menstrual cycle after the divorce, or by giving birth if she is pregnant, or by the passing of three months if she does not menstruate. Know that your wife is not obligated to care for your mother or serve her, and you do not have the right to compel her to do so. However, if she volunteers to do so, it is better and brings greater reward.