Carrier-based jets and warships shot down 21 drones and missiles launched by the Iran-backed group on Tuesday night.
The UN Security Council passed a resolution on Wednesday demanding an immediate end to the Houthi attacks.
The text endorsed the right of UN member states to defend their vessels. The Houthis reacted scornfully to it.
Their spokesman Mohammed Ali al-Houthi called the resolution a "political game". They claim to be targeting Israeli-linked vessels, in protest at Israel's war against Hamas in Gaza.
The UN resolution demanded "that the Houthis immediately cease all such attacks, which impede global commerce and undermine navigational rights and freedoms as well as regional peace and security". Eleven nations voted for it, but Russia, China, Mozambique and Algeria abstained.
Earlier, the US and several allies warned of "consequences" for the Houthi attacks in the Red Sea. Asked about potential strikes in Yemen, UK Defence Secretary Grant Shapps said: "Watch this space."
The International Chamber of Shipping says 20% of the world's container ships are now avoiding the Red Sea and using the much longer route around the southern tip of Africa instead.
The Houthis said they targeted a US ship on Tuesday providing support to Israel. It was the 26th attack on commercial shipping in the Red Sea since 19 November.The US military said Iranian-designed one-way attack drones, anti-ship cruise missiles and anti-ship ballistic missiles were launched from Houthi-controlled areas of Yemen at around 21:15 local time (18:15 GMT).
Eighteen drones, two cruise missiles and one ballistic missile were shot down by F/A-18 warplanes from the aircraft carrier USS Dwight D Eisenhower, which is deployed in the Red Sea, and by four destroyers, the USS Gravely, USS Laboon, USS Mason and HMS Diamond.
HMS Diamond shot down seven of the Houthi drones using its guns and Sea Viper missiles, each costing more than £1m ($1.3m), a defence source said.
No injuries or damage were reported.
Later, Houthi military spokesman Yahya al-Sarea confirmed its forces had carried out an operation involving "a large number of ballistic and naval missiles and drones".
"It targeted a US ship that was providing support for the Zionist entity [Israel]," he said.
"The operation came as an initial response to the treacherous assault on our naval forces by the US enemy forces," he added, referring to
He added that the rebels would "not hesitate to adequately deal with all hostile threats as part of the legitimate right to defend our country, people and nation".
Mr Sarea also reiterated that the Houthis would continue to "prevent Israeli ships or ships heading towards occupied Palestine from navigating in both the Red Sea and the Arabian Sea until the [Israeli] aggression [on Gaza] has come to an end and the blockade has been lifted".
A spokesperson for UN Secretary General Antonio Guterres said he was "very concerned" because of the risks the situation posed to global trade, the environment and lives, as well as the "risk of the escalation of the broader conflict in the Middle East".
IMAGE SOURCE,PA MEDIAImage caption,
The UK's HMS Diamond and three US warships helped shoot down the Houthi drones and missiles
Mr Shapps warned on Wednesday that the UK and its allies had "previously made clear that these illegal attacks are completely unacceptable and if continued the Houthis will bear the consequences".
"We will take the action needed to protect innocent lives and the global economy," he added.
Later, the defence secretary said in a TV interview that Iran was "behind so much of the bad things happening in the region" and warned the Islamic Republic and the Houthis that there would be "consequences" if the attacks on shipping did not stop.
Asked if there could be Western military action against Houthi targets in Yemen, or even targets inside Iran, he replied: "I can't go into details but can say the joint statement we issued set out a very clear path that if this doesn't stop then action will be taken. So, I'm afraid the simplest thing to say [is] 'watch this space'."
He was referring to a statement put out a week ago by the UK, US, Australia, Bahrain, Belgium, Canada, Denmark, Germany, Italy, Japan, Netherlands, New Zealand, South Korea and Singapore, who launched "Operation Prosperity Guardian" last month to protect Red Sea shipping.
They said the attacks posed "a direct threat to the freedom of navigation that serves as the bedrock of global trade in one of the world's most critical waterways".
It may not have had the bravado of Mr Shapps' "watch this space" warning, but US Secretary of State Antony Blinken was also clear in his condemnation of the incident.
Speaking to reporters at an airport in Bahrain during a Middle East tour, he was pressed by BBC North America correspondent Anthony Zurcher about whether it was time that talk of consequences turned to US action.
Mr Blinken responded that he did not want to "telegraph" a US military move, but that he had spent the past four days in the region warning the Houthis to cease their aggression.
They have not only refused, but after this latest strike have claimed they are specifically targeting US ships.
Almost 15% of global seaborne trade passes through the Red Sea, which is linked to the Mediterranean by the Suez canal and is the shortest shipping route between Europe and Asia.
The fear is that fuel prices will rise and supply chains will be damaged.
The Houthis say they have been targeting Israeli-owned or Israel-bound vessels to show their support for the Iran-backed Palestinian group Hamas since the start of the war in Gaza in October.
Formally known as the Ansar Allah (Partisans of God), the Houthis began as a movement that championed Yemen's Zaidi Shia Muslim minority.
In 2014, they took control of the capital, Sanaa, and seized large parts of western Yemen the following year, prompting a Saudi-led coalition to intervene in support of the international-recognised Yemeni government.
The ensuing war has reportedly killed more than 150,000 people and left 21 million others in need of humanitarian assistance.
Saudi Arabia and the US have accused Iran of smuggling weapons, including drones and cruise and ballistic missiles, to the Houthis in violation of a UN arms embargo. Iran has denied the allegation.