How does Carver use irony to convey the theme in "Popular Mechanics"?

This story alludes to the story of King Solomon's judgment in the Biblical book of Kings. In this story, two women claim to be the mother of one baby. One woman says that the other woman rolled over on her own child while she was sleeping and then stole the first woman's child to replace her own. The women argue the point, and Solomon orders that a sword be brought to cut the baby in half; then each woman can have a part of him. The baby's mother begged Solomon not to kill him, even if it means that the other woman gets to have him, while the other woman says to cut go ahead him in half so that neither one can have the baby. In this way, Solomon learns who the baby's real mother is, because she cares more that he lives than that she gets to keep him, and so Solomon awards the child to her.
Likewise, in Carver's story, two parents fight over a baby to whom they both lay claim. As their relationship disintegrates, the husband and wife begin to physically pull and tug on the child's body. She says that he's hurting the baby, and he denies it. The child is screaming, likely in both fear and pain.

She would have it, this baby. She grabbed for the baby's other arm. She caught the baby around the wrist and leaned back. But he would not let go. He felt the baby slipping out of his hands and he pulled back very hard.

Each parent is only thinking about what they want, and neither seems to truly consider the safety of the child. It is ironic that individuals who claim to care about something would seem to rather destroy it than allow another to have it. We would expect one person—the one who cares more about the child—to relent, like the woman in the Solomon story. However, unlike the Solomon story, neither party would prefer to see the child safe with the other partner, and so we can interpret the final line as a reference to the child's death (the fate it would have had in Kings had the real mother not relented in order to guarantee the child's life).

Both the irony and the theme of "Popular Mechanics" are quite dark.  A man is packing his clothing, and there is a framed picture of the baby that he intends to take with him; it is on the bed, and the woman selfishly takes it away from him.  It is clear that their relationship is ending; she tells him, "I'm so glad you're leaving."  
When the man announces his intention to take the baby with him, they continue to argue.  She is holding the baby, and the baby begins to cry at the sound of their angry voices.  Instead of stopping, like two responsible and caring adults, the man and woman both grip the baby, who begins to scream. It is ironic that two people who are fighting over who will keep the baby then emotionally and physically harm the baby.  Ultimately, they hurt the baby in their struggle, and neither one gets him.  Either they have hurt him badly enough that neither will be allowed custody, or worse, they have fatally injured him.
The theme of this very dark story has to do with immaturity, selfishness, and inhumanity between two people in a relationship. Neither the mother nor the father is fit to have a child; they are clearly emotionally immature and selfish, uncivilly grappling over a helpless baby.