After a big quarrel, how can the relationship be properly repaired?

Repair needs three things: acknowledging the mistake specifically, putting right its effects, and changing behaviour so it does not happen again.
Allah praises those who, when they do wrong, quickly remember Allah and do not persist in it (Al ʿImran 3:135), and the Prophet ﷺ advised following a bad deed with a good one to wipe it out (Jamiʿ at-Tirmidhi 1987). Steps: wait until both are calm; apologise without "but"; listen to how your spouse was hurt; agree on one concrete change; and do small things that rebuild a sense of safety, such as touch, kind words and time together. Avoid covering a quarrel with gifts without discussing its cause. If the quarrel involved violence, repair begins with safety and professional help, not just an apology. See Persuasion techniques.