Abiy has rejected international "interference" in the conflict even as the UN and others pleaded for weeks for unhindered access to Tigray as food, medicines and other supplies ran out.
Now Covid-19 has emerged as the latest source of alarm.
"Only five out of 40 hospitals in Tigray are physically accessible," the new UN report issued on Thursday says.
"Apart from those in [the Tigray capital] Mekele, the remaining hospitals are looted and many reportedly destroyed."
It does not say who did the looting.
Covid-19 surveillance and control work was interrupted for more than a month in Tigray, and that, along with the displacement of hundreds of thousands of people, "is feared to have facilitated massive community transmission of the pandemic", the report says.
Ethiopia has one of the highest Covid-19 caseloads on the African continent with more than 127,000 confirmed infections. While its rate of daily cases has declined in recent weeks, officials have not said whether they have been receiving any data from the Tigray region.
"Health facilities outside of major cities are nonfunctional and those in the major cities are partially working with limited to no stock of supplies and absence of health workers," the UN report says.
The report also says the Tigray region remains volatile.
"Localised fighting and insecurity continues, with fighting reported in rural areas and in the peripheries of Mekele, Shiraro and Shire among other locations, as of last week," it says.
The overall humanitarian situation is "dire," the U.N. says, with food supplies "very limited" and widespread looting reported.
"Only locally produced food items are available and at increasing prices, making basic goods unaffordable."
Most Tigray residents are subsistence farmers, and the conflict disrupted the harvest.
Two important camps hosting tens of thousands of refugees from nearby Eritrea remain unreachable — another source of alarm as the presence of Eritrean troops has been confirmed in Tigray.
No one knows how many thousands of people have been killed in the conflict. At least five humanitarian workers have been killed.