Manhattan, Brooklyn, Riverdale and Westchester have many diocesean Catholic schools governed and financed by the Catholic dioceses.
Many New York families seek admission into independent Catholic schools. Independent Catholic schools are funded and governed independently--not by the dioceses.
In Manhattan, families who work with me often seek admission into Saint David's for boys, and Marymount and Sacred Heart for girls. All three schools accept students of all religions. From Marymount's website:
While committed to its Catholic heritage, Marymount School respects the religious diversity of the community and seeks to give all students a deeper understanding of the role of the spiritual in life, an appreciation of individual religious traditions, and a commitment to service.