Lise Grande, UN humanitarian coordinator for Yemen, earlier said hundreds of thousands of people depend on Al-Thawra, the country's largest hospital.
United States has been providing logistical support to the Saudi-led coalition to carry out attacks on the Yemen soil, since March 2015.
However, a coalition spokesman said the Houthi militia was behind the deaths of civilians in Hudaida, according to Al Arabiya TV.
Griffiths is planning to invite the two sides to Geneva in September for consultations aimed at convening peace talks, to "discuss a framework of negotiations and specific plans for moving the process forward".
Saudi Arabia suspended oil exports through the channel following last week's Houthi attacks on crude tankers.
The Saudi-led coalition, backing Yemen's internationally recognized government, seeks to retake rebel-held areas along the western coast, including Hodeida, a vital lifeline for most Yemenis.
The UN had been trying to broker a deal to avert an attack on Hodeidah by the Saudi alliance due to fears it would further hinder Yemenis' access to food, fuel and medicine; worsening the world's most urgent humanitarian crisis.
Britain's UN Ambassador Karen Pierce, the current council president, said after the meeting that "there was very strong support" for Griffiths and "we're united in his efforts to get the parties to Geneva".
The Houthis, allied with Saudi Arabia's regional arch-rival Iran, control Yemen's capital Sanaa and Hodeida port, the entry point for around three quarters of the impoverished country's imports.
"There is a pond of blood outside the hospital's building", he said. The Saudi-led coalition of mostly Persian Gulf nations has been carrying out airstrikes against the Houthis at Hadi's request since March 2015.
"The scenes coming from Hodeida are horrific".
"You have heard the statistics many times: more than 22 million people - 75 per cent of the population - require humanitarian assistance and protection".
"Civilians are at risk, infrastructure is at risk, and we as the global community have to demand that the two parties come together and understand the seriousness of this", said Haley.