By actually listening to her, he found that she doesn't have an intractable philosophical opposition to abortion. Her opposition to abortion is rooted in her insecurity because her husband left her because she's barren; she feels like she's failed at being a woman, so other women getting abortions feels like a spiteful act to her.
Clearly, there is in fact room for compromise with her. At that point, the author could probably have shared his story about women he knew who'd had abortions, for whom it was a painful decision that wasn't thoughtless and casual and spiteful, and wasn't an indictment of her worth. Done carefully, she probably would have acknowledged that it's properly a deeply personal decision and she's not really that committed to her own feelings.
We do know that she doesn't have an intractable philosophical opposition because she acknowledges that her opposition is based on her circumstances. That's neither philosophical nor intractable. "Intractable" would be something like "I just believe it's totally wrong and won't entertain discussion otherwise."
Room for compromise comes from the fairly obvious approach of disassociating abortion from her personal circumstances... likely by sharing the author's stories about women he knows for whom abortion was a difficult but justified personal choice, not a spiteful act directed at the woman to shame her for being barren. With some empathetic discussion, this doesn't seem an unreasonable outcome. What more do you want, to believe that there is room for compromise?
Clearly, there is in fact room for compromise with her. At that point, the author could probably have shared his story about women he knew who'd had abortions, for whom it was a painful decision that wasn't thoughtless and casual and spiteful, and wasn't an indictment of her worth. Done carefully, she probably would have acknowledged that it's properly a deeply personal decision and she's not really that committed to her own feelings.