I agree with the OP, people should take responsibility for their actions and it should be flat illegal to abandon a child without first finding the child a suitable family who is willing to take legal custody.
Besides I will never understand the failed logic that leads to people believing that it is more humane to abandon a child than it is to have an abortion ensuring the child will never ever suffer at all EVER nor will it ever want for anything.
On a side note I think if either parent is to declare that they don't want a child before it is born that abortion should be mandatory however if the child is born it should be mandatory that the parents stay together and raise it properly because they lost the right to choose what is best for them when they committed to doing what is best for the child by having it in the first place.
This is one area where I think the government needs to step in and lay down some rules even if it does step on some religious beliefs as your beliefs no longer matter when the well-being of a innocent child is on the line because of your own actions.
Wouldn't the father be able to claim the baby and sue the mother for child support in this case?
I thought giving up the child is only OK if both parties agree.
You are working under the assumption the father knows of the child's existance. If the mother is planning to give up the child it is far more likely that either she doen't know who the father is or at the very least has no intention of telling the father.