I suspect it's closer toward "realism," because the paint is applied in solid colors and the painting is not so much a study of the effects of light as impressionist works are. The outlines are more distinct than those in works by Monet or Renoir, two famous impressionists. However, it isn't exactly photo realism. The brush strokes feel light and flow more freely than realist works do. It has more "atmosphere" and spontanaeity than works of the "realist" school. Neither of the two alternative classifications you offer seem immediately applicable. The technique derives from Frans Hals.
What a lovely picture! --the sheen of the dress and the gauzy neckpiece are perfectly captured, and the larger setting is suggested so easily by the rug and the settee. The subject of the painting, Mrs. Hammersley herself, seems to have just looked up. Sargent has captured her spirit. The whole seems to have painted with a verve and sympathy immediately captivating--a superior society portrait.