The story is written from Mr. Waythorn's perspective, and Edith Wharton makes it clear that he is an exceptionally poor analyst of human personality. He has apparently given little thought to what it would mean to marry a divorced woman, or to what had motivated his wife, Alice, to choose him. Waythorn is so confident of his own worth and so invested in the idea of being the "possessor" of a wife that he has seriously underestimated Alice.
Waythorn does not see Alice as stubborn. The third-person narrator describes his image of his wife as being pliant, so easily adaptable that he has started to feel sick over it. When confronted with her ex-husbands in the flesh, finally Waythorn realizes that she had sex with them and his pride in possession starts to wane because it is not his unique entitlement. Rather than stubborn, he calls her elastic.
She was "as easy as an old shoe"—a shoe that too many feet had worn. Her elasticity was the result of tension in too many different directions.
We might better characterize her as tenacious. Alice has persisted with her involvement in the institution of matrimony through two previous marriages that proved unsuccessful in one way or another. She has obtained a third husband who seems solid and unimaginative and has made him devoted to her and accepting of her daughter, Lily. While less tenacious women might have given up on marriage, Alice has persevered. One could say she has been stubborn in finding the marriage that works for her.