Is there something in Scripture about when a father walks away from his child?

Not that I can think of. The Bible gives guidelines for the ideal relationship between parents and children. For example, in Ephesians 6:1-4 it says

Children, obey your parents in the Lord, for this is right. Honour your father and mother (this is the first commandment with a promise), that it may go well with you and that you may live long in the land. Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord.

Some of the most powerful lessons on being a father are looking at how God is described as a father. For example, in Luke 11:13 he is described as a loving, generous father. In Hosea he is described as dealing gently with his toddler son Israel (also called Ephraim in this passage):

When Israel was a child, I loved him,
and out of Egypt I called my son.

But the more they were called,
the more they went away from me.
They sacrificed to the Baals
and they burned incense to images.

It was I who taught Ephraim to walk,
taking them by the arms;
but they did not realize
it was I who healed them.

I led them with cords of human kindness,
with ties of love.
To them I was like one who lifts
a little child to the cheek,
and I bent down to feed them.    (Hosea 11:1-4 TNIV)

In this passage, God is lamenting his wayward son and remembering the times when he comforted and taught him.

The parable of the lost son (Luke 15:11-32) is similar. There the father did not give up on his rebellious son, but was ready to welcome him home when he repented. In the same way, despite the rebellion of his children, God does not give up on us, but is ready to take us back when we repent.