He later walked back his comments after Mr Gutwein said contact tracing had not found evidence of the party.
However, the party claim is being investigated by state police.
Mr Gutwein delivered a blunt message to people criticising hospital staff on Facebook and urged northwest community members to work together.
"Anyone that's thinking about lining up a health care worker, taking a pot shot at them, should just draw breath and accept there is a process underway," he said on Wednesday.
"We will work through it. We will get to the bottom of it, and then we will explain fully what has happened."
Of Tasmania's 169 coronavirus cases, more than 80 are linked to the outbreak at the North West Regional Hospital and North West Private Hospital in Burnie.
More than 45 health workers have tested positive to the deadly virus, with a spike in cases over the Easter holidays.
Four new cases were confirmed in Tasmania on Wednesday, all in the northwest.
The state's sixth death, a 91-year-old woman at the Mersey Community Hospital at Latrobe in the northwest, was linked to the outbreak.
The state government will begin a "testing blitz" in the region and has brought in additional laboratory technology to boost capacity.
"If you think it's just a sore throat, think about your community," Health Minister Sarah Courtney said, noting that up to 500 people per day can be tested.
"Make sure if you're experiencing symptoms, call the public health hotline or your GP."
Testing has been offered to staff of the two hospitals, which are both being deep cleaned by specialist teams.
It is hoped the facilities' emergency department will reopen on Thursday.
Medicos from the Australian Defence Force and Australian Medical Assistance Teams, usually reserved for national disaster relief, will help get services back online.
The hospital closures have forced 1200 staff and their families into quarantine, about 5000 people in total.
AAP