"To the woman's family, friends, and loved ones - I'm sorry. I extend our condolences and sympathies to you all," he said.
Mr Gutwein announced that tough retail restrictions in the state's northwest, where an outbreak has shut two hospitals, would be extended for another week.
He also said testing for coronavirus would be ramped up in the area.
"We need to crush this virus," Mr Gutwein said.
Anyone "who has even the slightest sniffle" should be tested, he added.
There have been 205 confirmed virus cases in Tasmania, with 130 of those in the northwest. For the first time this month, the state on Thursday recorded no new virus cases.
NSW on Friday also announced plans to ramp up its COVID-19 testing, with plans to check more than 8,000 people a day.
Premier Gladys Berejiklian said anyone with symptoms, or is concerned they have COVID-19, should come forward and get tested.
"Anybody across the state, if you have symptoms, if you're worried you have COVID-19, please come forward and get tested," she told reporters in Sydney.
"We want to see the number of tests go up above 8,000 a day. As we consider easing restrictions, we need to have more tests."
More than 7,300 tests were completed on Thursday with seven new cases of COVID-19 reported across the state. The total number of confirmed cases in NSW is 2,982.
It comes as the state's COVID-19 death toll reached 34 after a fourth resident at Anglicare's Newmarch House in Caddens died.
NSW Health confirmed the 79-year-old woman died early on Thursday morning.
Some 44 people at the western Sydney facility - 29 residents and 15 staff - have been infected with COVID-19.
Outbreak at Victorian psychiatric facility
Meanwhile, authorities in Victoria are investigating an outbreak of coronavirus at a psychiatric facility.
Health Minister Jenny Mikakos confirmed there have been 14 known cases linked to the outbreak at the inpatient facility, which she did not name.
"Further testing will be undertaken today to determine if there is any further spread at the facility," she told reporters on Friday.
The first confirmed case of COVID-19 at the facility was in late March, with five patients, five staff and four known contacts having now contracted the virus.
Three of them are in hospital, with one in intensive care.
Victorian Chief Health Officer Brett Sutton said the facility had been shut down and activities such as group therapy suspended.
"We're doing very substantial things but we understand the explosive potential in enclosed facilities, as we've seen in hospitals and aged care facilities," he said.
Victoria recorded six new cases of coronavirus overnight, bringing the state's total to 1343. About 1250 people have recovered.
Twenty-seven people are in hospital with the virus, including 11 in intensive care.
"This reinforces the point that this is not all over," Ms Mikakos said.
People in Australia must stay at least 1.5 metres away from others and gatherings are limited to two people unless you are with your family or household.
If you believe you may have contracted the virus, call your doctor (don’t visit) or contact the national Coronavirus Health Information Hotline on 1800 020 080. If you are struggling to breathe or experiencing a medical emergency, call 000.
SBS is committed to informing Australia’s diverse communities about the latest COVID-19 developments. News and information is available in 63 languages at sbs.com.au/coronavirus.