The city's medical examiner said 52-year-old Cornell — the lead singer in Soundgarden and Audioslave — died early Thursday morning after performing at a concert on Wednesday night.
But Cornell's family said without toxicology test results completed they do not know what caused his death.
On Friday, an official said the results of a full autopsy and toxicology tests could take days before they are completed and released.
Cornell's wife Vicky said when she spoke to her husband after the Detroit show, he slurred his words and told her he may have taken "an extra Ativan or two".
She said "he was different", and she contacted security to check on him.
"What happened is inexplicable, and I am hopeful that further medical reports will provide additional details," she said.
"I know that he loved our children and he would not hurt them by intentionally taking his own life."
According to lawyer Kirk Pasich, the musician had a prescription for anti-anxiety drug Ativan.
The sedative has side effects that can include drowsiness and dizziness, according to the National Institutes of Health.
"Without the results of toxicology tests, we do not know what was going on with Chris — or if any substances contributed to his demise," Mr Pasich said in a statement.
"Chris, a recovering addict, had a prescription for Ativan and may have taken more Ativan than recommended dosages.
"The family believes that if Chris took his life, he did not know what he was doing, and that drugs or other substances may have affected his actions."
Cornell had history of addiction
Cornell had battled addiction problems in the past.
He told Rolling Stone in a 1994 interview he started using drugs at age 13, and was kicked out of school at 15.
"I went from being a daily drug user at 13 to having bad drug experiences and quitting drugs by the time I was 14, and then not having any friends until the time I was 16," he said.
"There was about two years where I was more or less agoraphobic and didn't deal with anybody, didn't talk to anybody, didn't have any friends at all.
"All the friends that I had were still [messed] up with drugs and were people that I didn't really have anything in common with."
Cornell was a leader of the grunge movement with Seattle-based Soundgarden — with whom he gained critical and commercial acclaim.
He also found success outside the band with other projects, including Audioslave, Temple of the Dog and solo albums.
Soundgarden's current tour kicked off in late April and was planned to run through May 27.
Grief-stricken fans left flowers at memorials across Seattle for the musician whose forceful, sombre songs helped cement the city's place in rock history.
One of the locations where people gathered was the Sound Garden art sculpture at a Seattle park, for which Soundgarden was named.
The city's Space Needle went dark at 9:00pm for an hour in tribute to Cornell.
He was born and raised in the city and was part of a close-knit group of artists who formed the foundation of what would become the grunge scene that exploded in the early 1990s, by combining the bombast of early 1970s heavy metal with the aggression and attitude of punk rock.