Is it necessary to search for the owner of the car who caused the accident, after he refused the repair estimate and requested to go to the workshop, and then has not made contact until now?

If you are able to find the man and determine the value of the damage, then you must pay it to him. If you are unable to reach him, then give charity on his behalf with an amount that absolves you. Since you gave him your number, he might contact you, so do not rush to give charity on his behalf. This is because if you give charity and he is only satisfied with direct payment, then you are obliged to pay, and the reward for the charity will be yours.