Is the justice system fair to men when it comes to custody?

When a mother gives birth if she is not married she is considered the custodial parent and the man has to file for his rights which in many cases are denied or are not as equal as those of the mother. Is this fair considering both he and the mother went half on the baby they concieved? What about in instances where a male was not ready to be a parent. Should he be allowed to request to sign over his rights and not be financially obligated to that child? From my understanding if a mother is not in agreeance with him signing over his rights then he is obligated to pay child support.