Following the crash in the Mar Vista area of Los Angeles, the 53-year-old actor was taken to hospital, where she is being kept on life support to see if she is a potential match for organ donation.
Under California law, however, she is legally dead.
"As of today, there will be no further investigative efforts made in this case," the Los Angeles Police Department (LAPD) said.
"Any information or records that have been requested prior to this turn of events will still be collected as they arrive as a matter of formalities and included in the overall case. When a person suspected of a crime expires, we do not present for filing consideration."
The department previously confirmed to the PA news agency that the crash was being investigated as a felony driving-under-the-influence (DUI) traffic collision.
An LAPD spokesman earlier said preliminary blood tests had revealed the presence of drugs in Heche's system but additional testing was required to rule out any substances that were administered in the hospital.
Pictures and video footage obtained by US media outlet TMZ following the incident showed Heche driving a blue Mini Cooper, which was later pictured severely damaged at the scene.
The vehicle struck a two-storey home and "erupted in heavy fire", according to the Los Angeles Fire Department (LAFD).
Fifty-nine firefighters took 65 minutes to access, confine and fully extinguish the flames within the heavily damaged structure, the LAFD said.
Heche suffered a "significant pulmonary injury requiring mechanical ventilation" and "burns that require surgical intervention", a spokeswoman for the actor previously confirmed to PA.
Heche is known for films including Donnie Brasco, Six Days, Seven Nights, Cedar Rapids and the 1998 Psycho remake.
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