According to legal docs, obtained by TMZ, Petito's family blames the Moab City PD for Gabby's death ... claiming Brian killed Gabby shortly after the department "failed to adequately respond to reports and evidence of domestic violence between Brian and Gabby."

The docs go on to recount the events leading up to her death ... including when police interviewed them on the side of the road last year -- with one officer saying she was crying uncontrollably and had cuts on her face and arm.

As you know, Brian told officers Gabby had been under emotional distress, allegedly pushing her away as she tried to slap him.

But, the family says there were inconsistencies in Brian's story -- including claiming he wanted to walk away from Gabby despite a witness alleging he tried to drive off without her.

Gabby's family claims the police didn't do enough probing of the situation to save Gabby -- adding that an independent investigator later concluded the officers "made several mistakes and could not rule out that Gabby's murder might have been prevented if the officers had handled the situation properly."

Laundrie family attorney, Steven Bertolino, tells TMZ, "The Moab police, in my view, did not contribute to the death of Gabby Petito in any way. My understanding of the Moab incident is that Gabby was the aggressor and admitted on camera to hitting Brian first. The Moab police report indicated that they could have arrested Gabby but chose not to and instead separated Brian and Gabby."

He continues, "The intervening days and events from the Moab incident to the date of Gabby's death appear to be far enough removed from the reasonable actions of the officers on the scene. I see no legal liability but maybe the City of Moab will settle the matter. So best of luck to the Petito family in their quest to recover for their loss from another source."