Mr Zeid expressed shock at a coalition air strike on a camp for internally displaced people in northern Yemen on Monday, which left at least 19 dead.
He also condemned attacks by rebels and allied army units on three hospitals.
The UN says at least 93 civilians have been killed since the 10-nation coalition launched an air campaign five days ago.
The coalition says it intends to "protect and defend the legitimate government" of President Abdrabbuh Mansour Hadi.
Mr Hadi fled abroad last week after the Zaidi Shia rebels advanced on the city of Aden, where he had taken refuge after the Houthis took full control of the capital Sanaa in January and placed him under house arrest.
Aid flight 'prevented'
Mr Zeid said the situation in Yemen was "extremely alarming" and warned: "The country seems to be on the verge of total collapse."
The UN statement said that there were differing accounts as to how many people were killed in Monday's air strike on al-Mazrak camp, but that UN staff had verified at least 19 fatalities. The camp, in the Houthis' heartland of Saada province, is home to some 4,000 people.
"The killing of so many innocent civilians is simply unacceptable," Mr Zeid said. "The principles of proportionality, distinction, and precaution fully apply in this context. International human rights law and humanitarian law should be fully respected."
Mr Zeid also stressed that medical facilities had a special protected status under international law amid reports that the Yemeni army's 33rd Armoured Division and Houthi-affiliated brigades had attacked three hospitals, causing an unknown number of casualties.
Heavy fighting has also been reported in the streets of Aden, where rebel forces are battling militiamen and Sunni tribesmen loyal to President Hadi.
The high commissioner said that in addition to the civilian casualties, homes, hospitals, education facilities and infrastructure in several locations had been destroyed, "making life even more difficult for the people in this war-torn country".
The International Committee of the Red Cross (ICRC) also expressed concern at the high number of civilian casualties and called for the urgent removal of obstacles to the delivery of vital medical supplies.
In a statement, the agency said a shipment was due to arrive in Sanaa by plane on Tuesday for distribution at hospitals across the country, but that efforts to negotiate the safe arrival of the plane with the Saudi-led coalition "have not been successful".
"In order that the wounded get the treatment they deserve, it's essential we deliver urgent medicines and surgical kits," said Cedric Schweizer, the head of the ICRC's operation in Yemen.
Meanwhile, the Saudi ambassador to the UK told the BBC that his country's regional rival, Iran, had been offering the Houthis military assistance, something that the rebels and the Iranians deny.