Jew is not necessarily a genuine one either. N ow to deal with the question that the reader posed. Specifically in the case of your inquiry, most of the rabbis would consider him still eligible to receive many of the privileges afforded to physical Jews, however I m not too sure HaShem and his Torah would feel the same way. The full scope of being a loyal Jew includes involvement in the Jewish community and support for Isra el. Historically, when the Jews became disinterested, by faith, with HaShem, then he became equally disinterested with them.
Covenants require a response on the part of the participants.