Question:Is there a particular scripture that tells that Bathsheba was an Israelite?
Bathsheba was married to a Hittite, but we can surmise that she was an Israelite because her father's name was Eliam (II Samuel 11:3) or Ammiel (I Chronicles 3:5). These names are Hebrew. Eliam means "God of the people" and Ammiel means "people of God." There was an Eliam among David's mighty men (II Samuel 23:34) and some wonder if Bathsheba was his daughter. If so they are from the city of Giloh, believed to be a suburb on the south side of Jerusalem.
There was an Ammiel among the sons of Korah who served as a gatekeeper in the Temple. If Bathsheba was his daughter, then she would have been a Levite. There was also an Ammiel who had a son named Machir from Lo Debar, a town just south of the Sea of Galilee on the eastern side of the Jordan. I suspect that this Ammiel was too old to be Bathsheba's father.
Of course, it is also possible that none of these men was the father of Bathsheba. But we do know that he carried a Hebrew name, so he was very likely an Israelite. Bathsheba's name also has meaning in Hebrew, it means "daughter of an oath."