The Saudi-led coalition did not immediately acknowledge the incident.
Robert Mardini, the regional director for the Red Cross, said the push on Hodeida "is likely to exacerbate an already catastrophic humanitarian situation in Yemen".
The offensive is controversial because the port serves as the entry point for 70 percent of Yemen's imports as the country teeters on the brink of starvation.
The conflict in Yemen has raged since late 2014, when the Houthis and allied forces seized north-western parts of the country, including the capital Sanaa, and eventually forced President Abdrabbuh Mansour Hadi to flee overseas. The sound of heavy, sustained gunfire clearly could be heard in the background.
Yemen's government declared Tuesday night that negotiations had failed to force the rebels from Hodeida, and that a grace period for UN-led peace efforts was over.
The Security Council has strongly supported efforts by new United Nations special envoy Martin Griffiths to resume political negotiations and avoid a military escalation of the three-year-long conflict that has killed more than 10,000 people, displaced more than 2 million, and created the world's worst humanitarian crisis. The Saudi-led coalition entered the war in March 2015 and has received logistical support from the U.S. Saudi-led airstrikes have killed large numbers of civilians and damaged vital infrastructure.
The United Nations said it was engaged in "intense" shuttle diplomacy between the Houthis and coalition leaders Saudi Arabia and the UAE to avert the attack.
The UAE warship was hit on Wednesday shortly after Saudi-led troops began a major operation to bring Hudaydah under their control, Abdulsalam Jaber, the spokesman of the national salvation government, said.
"Under global humanitarian law, parties to the conflict have to do everything possible to protect civilians and ensure they have access to the assistance they need to survive", Lise Grande, U.N. humanitarian coordinator for Yemen, told Reuters by email.
"We are in constant contact with all the parties involved to negotiate arrangements for Hodeidah that would address political, humanitarian, security concerns of all concerned parties", he said.
Defense Secretary Jim Mattis on Monday acknowledged the United States continues to provide support to the Saudi-led coalition. "We have a ship offloading food even as shelling and bombing is happening", Grande said.
The assault began after the expiry of a deadline set by the United Arab Emirates for the Houthis, who hold the capital Sanaa, to hand over the sole port under their control.
The liberation of Hodeidah port will also cut supplies of the Iranian weapons, used against the Yemeni people, expand the legitimacy- liberated areas, enable resumption of secure navigation and dries up smuggling sources for the Iranian-backed militias.
The Houthis deployed military vehicles and troops in the city centre and near the port, as warplanes struck the coast to the south, a resident speaking on condition of anonymity told Reuters.
The Houthis, with roots in a Zaidi Shi'ite minority that ruled a thousand-year kingdom in Yemen until 1962, deny they are pawns of Iran.
Yemen lies beside the southern mouth of the Red Sea, one of the world's most important trade routes, where oil tankers pass from the Middle East through the Suez Canal to Europe.
The UAE's Gargash said plans were in place to keep the port operational and urged the global community to pressure the Houthis to evacuate and leave the port intact, without planting mines. The accusations are denied by the group and Iran.