"Dr. Stone. Your patient," she said to the man who everyone believed to be my father, putting in his hands not only the life of a woman that he chose to love, but our two lives—mine and my brother's—which he chose to hate. (1.2.65)
When Matron she sees what's going on between Sister Mary's legs she abandons ship, handing over the wheel to Dr. Stone, who is not quite the captain she hoped for. Not that we blame her, really—she's way out of her league here.
Hemlatha had established that the boys could move their limbs, neither of them was cockeyed, and they seemed to hear and to see. "Thomas," she said, approaching, but he cringed. He turned away. He would not look. (1.10.36)
Stone seems to feel a physical repulsion to the babies, though they're his own flesh and blood. He "cringes," "turns away," and "won't look." It's like people who are afraid to watch a horror movie and won't peek at the screen even though their friends beg them to. Is Stone afraid of the children and what they represent?
Stone thrust his chin at her, as if to say she could name them whatever she wanted. "Please get them out of my sight," he said very softly. (1.10.43)
What a juxtaposition: the man's body language (chin jutting out) and words ("get them out of my sight") clash wildly with his tone (very soft). Why? How could he so calmly choose to abandon his children? The chin-jut, though, indicates that he's leaving Hema in charge, so maybe the knowledge that she'll care for them makes it easier for him to leave.
He kept his back turned on the infants to gaze once more at Sister Mary Joseph Praise. (1.10.45)
Aha! Now we're starting to peel back some of the layers of the onion that is Dr. Thomas Stone. It's not necessarily that he's abandoning his children (okay, yes, it is) but that he can't abandon Sister Mary. He can't accept her death, and that she, in a sense, has abandoned him.
Stone wanted to run away, but not from the children or from responsibility. It was the mystery, the impossibility of their existence that made him turn his back on the infants. (1.10.45)
Here we get some more clarification as to Stone's motivations for abandoning Shiva and Marion. He doesn't know how Sister Mary Joseph Praise got pregnant, though he's a doctor, so we're pretty sure he knows where babies come from. It turns out he and Sister Mary Joseph Praise had sex while he was in a drunken fever dream, so he didn't even realize that it had really happened. So romantic.
Sister Mary Joseph Praise lay lifeless and unburdened of the two lives she carried, as if that had been her sole earthly purpose. (1.10.46)
Sister Mary Joseph Praise dies on the operating table, just after her children have been delivered. She abandons her physical body and her babies all in one fell swoop. The description of her as "unburdened" makes her seem like a beast of burden, like a donkey or a mule, something meant only to carry things. The narrator even says that bringing the boys to earth was her only purpose.
"Stone, think about this," Hema said. "Turn your back on me if you want, because I'll have no use for you. But don't turn your back on these children. I won't ask you again."
[…] What she didn't see was any recognition of the infants as being connected to him. He spoke like a man who'd just been hit on the head. "Hema, I don't understand who… why they are here… why Mary is dead." (1.10.50-51)
Hema seems to think that all Stone wants is to abandon someone, and he'll be satisfied—so she offers up herself in exchange for the boys, who need their father. But Stone still doesn't know how Sister Mary Joseph Praise got pregnant; he might be thinking that she had a secret boyfriend (besides him). That makes it easier for him to abandon the babies.
At last he met her gaze, refusing to look down at the infants, and what he said wasn't what she wanted to hear. "Hema, I don't want to set eyes on them, ever." (1.10.53)
Good thing destiny had other plans for Thomas Stone, or we wouldn't have a novel to read. At this moment, Stone truly does reject the children and abandon them, knowingly. By stating that he doesn't ever want them in his sight, he is banishing them from his life. But they'll be back. Oh, they'll be back.
"You heard me, Stone, you killed her," Hema said, raising her voice so that she drowned out every other sound. He flinched as the words lashed into him. It pleased her. She felt no pity. Not for a man who wouldn't claim his children. He pushed the swinging door so hard it shrieked in protest.
"Stone, you killed her," she shouted after him. "These are your children." (1.10.61-62)
Hema doesn't even seem to be trying to bring Stone back at this point. Now she's just punishing him for abandoning his poor, defenseless sons just after their mother abandoned them by dying. Even the door seems to judge him, shrieking in protest as he walks out of the hospital, never to return.
And finally, reluctantly, almost as an afterthought, but because you cannot escape your destiny, and so that he wouldn't walk away scot-free, she added our surname, the name of the man who had left the room: Stone. (1.10.81)
It's a funny twist on Hema's part. She doesn't want Stone to come back for the babies, but she also wants to mark him as their father. She doesn't want to let him get away with abandoning them, because it was the wrong thing to do. It's also a sort of favor to the babies, connecting them to their father by name if by nothing else.