What is the ruling of Islamic law on a brother refusing to forgive his brother who sold his share of the car without his knowledge, and what are the consequences of that?

You did well by apologizing to your brother for what you did by selling a part of his car without his permission. Your brother's acceptance of his share of the price validates the sale, as it indicates his consent. However, the sin of transgression remains unless he forgives you. The advice is to repeat your apology to him, involve trustworthy mediators, increase good deeds, and show kindness to him, for indeed kindness removes resentment from hearts, in accordance with His, the Almighty’s, saying: "And not equal are the good deed and the bad. Repel [evil] with that which is better, and thereupon the one whom between you and him is enmity [will become] as though he was a devoted friend. But none is granted it except those who are patient, and none is granted it except one having a great portion [of good]."