I believe so, since all Jews are directly deccended from Abraham. Many orthodox Jews do not consider David a full Jew.
There is no such thing as a partial Jew.
David is Jewish. No Jew thinks otherwise.
First of all, Ruth was a Jew, and even if she wasn't, that would not affect David's status as a Jew. Ruth converted to Judaism before marrying Boaz and bearing Obed. See Ruth 1:16, where Ruth states her intention to convert. After Ruth converted, she was a Jew, and all of her children born after the conversion were Jewish as well. But even if Ruth were not Jewish at the time Obed was born, that would not affect King David's status as a Jew, because Ruth is an ancestor of David's father, not of David's mother, and David's Jewish status is determined by his mother.