Who is at fault and who should initiate the apology in a dispute between spouses caused by the husband's reprimand of his wife and accusation of negligence, due to their daughter returning alone from school?

You should have asked your wife about her excuse before accusing her of negligence. Your wife should have explained her excuse and not cut off communication. If it is proven that she was not negligent, then you must apologize to her for accusing her. This matter is simple and does not warrant dispute. The foundation of the marital relationship is compassion, understanding, and overlooking minor faults. Among the good manners of a wife is her striving to please her husband, and among the good manners of a husband is overlooking his wife's faults and treating her kindly.