Given Catherine Sloper's fate at the end of Washington Square, what is James trying to say about "New Woman"?
The novel offers a rather mixed view on contemporary life at the time. Catherine's development into a "modern woman" or into a resemblance of the "New Woman" is painful and isolating, but ultimately empowering.
However, we must point out that in the closing pages of the novel Catherine is aligned with values and modes of being that are out of date. She is seen to represent a type of person who no longer exists.
On several occasions, Catherine is compared to a wife of old; one who patiently waits, who forgives, who sees an explanation given by the husband as a favor and does not expect to get explanations often. She is decidedly not modern while she still hopes to marry Morris. In fact, she is quite explicitly aligned with the opposite type.
Yet, when she determines not to marry Morris, Catherine determines not to marry anyone. She maintains this stance and denies him when he makes his last call on her. Catherine, once in a powerless position regarding this man, now has power over him.
She has the power to say "no" and she exercises this power. Importantly, she had this power all along, but chose to exercise her passions instead of her economic position. At the story's end, Catherine is still not happy, but she is in control of her life. Her changes are significant, especially in light of Morris and Mrs. Penniman.
In the final chapter, it is revealed that only Catherine has changed in any measure.
While these other two characters are still drawn onto pursue their petty schemes and to achieve some dramatic success, Catherine has found a wholeness and an independence that allow her to enact what might be seen as a significant achievement of integrity.
Though Catherine is not a member of the social vanguard - not an especially "New Woman" - she lives in her old house with her old aunt, practicing her old ways with a quite new social power. She is not resigned to being a failure (at marriage, etc.) but is instead accepting. She does not need to marry.