The 27-year-old singer was found dead at her London home on Saturday.
Officials are now waiting for results of toxicology tests, which are expected in two to four weeks.
A coroner's official said police have found no suspicious circumstances in the death, but the cause of her death is being treated as "unexplained," rejecting speculation that she died from a drug overdose as "inappropriate."
Winehouse's parents formally identified her body on Monday and visited mourners outside her north London home to thank them for their support
"I can't tell you what this means to us - it really is making this a lot easier for us," said her father, Mitch Winehouse. "We're devastated and I'm speechless but thanks for coming."
A family spokesman said a private funeral "for family and close friends" would be held on Tuesday at an undisclosed time and place.
The Associated Press contributed to this report.