At times a husband-wife relationship reaches a stage in problems where reconciliation is difficult, and the only option is divorce.
In many cases children are also involved, and so the worse thing for all parties involved (husband, wife, and child/children) is, in addition to the difficulty of separation, that tension and further altercations are involved.
Unfortunately, we can see that in some cases things escalate so bad, that families are dragged to court, and it becomes a battle between two people who once loved each other.
This is fundamentally against what the Quran instructs us when divorce occurs (see: Surah al-Baqarah, verse 229). The couple should part away with benevolence and grace.
This also means that co-parenting is very important for their sake, and for the child's sake as well. Both parents must try to contribute in the best manner and bear in mind the interests of the child.
In this case, it is for the father's interest, and the child's interest that he maintains good ties with his ex-wife, and the mother of their child. He should financially assist as much as he can, and the mother's well-being will also add to the emotional stability of their child as well.
The father should be involved in the child's life, and the mother should not deprive him of that, nor in any way should they allow their misunderstandings or disputes to affect the quality of their co-parenting.
As for the shar'i side of the rights of custody, the details of these laws should be looked at within this framework of thinking, where the overall interests of all parties are taken into consideration and nobody is wronged.
It is always advisable to remain patient, wise, and consult with professionals and experienced people in how to deal with circumstances that may arise.
And Allah knows best.