The airstrikes on Saturday, which the Houthi-run health ministry said killed at least 53 people, are the biggest US military operation in the Middle East since President Donald Trump took office in January.
One US official told Reuters the campaign might continue for weeks.
Houthi leader Abdul Malik al-Houthi said on Sunday his militants would target US ships in the Red Sea as long as the US continues its attacks on Yemen.
"If they continue their aggression, we will continue the escalation," he said in a televised speech.
The Houthi movement's political bureau described the attacks as a "war crime", while Moscow urged Washington to end the strikes.
The Houthis' military spokesperson said, without offering evidence, the group had targeted US aircraft carrier USS Harry S Truman and its warships in the Red Sea with ballistic missiles and drones in response to the US attacks.
US warplanes shot down 11 Houthi drones on Sunday, none of which came close to the Truman, a US official told Reuters.
US forces also tracked a missile that splashed down off the coast of Yemen and was not deemed a threat, the official said.
US Defense Secretary Pete Hegseth told Fox News' Sunday Morning Futures: "The minute the Houthis say we'll stop shooting at your ships, we'll stop shooting at your drones. This campaign will end, but until then it will be unrelenting".
"This is about stopping the shooting at assets ... in that critical waterway, to reopen freedom of navigation, which is a core national interest of the United States, and Iran has been enabling the Houthis for far too long," he said.
"They better back off."
The Houthis, who have taken control of most of Yemen over the past decade, said last week they would resume attacks on Israeli ships passing through the Red Sea if Israel did not lift a block on aid entering Gaza.
They had launched scores of attacks on shipping after Israel's war with Hamas began in late 2023, saying they were acting in solidarity with Gaza's Palestinians.
Trump also told Iran, the Houthis' main backer, to stop supporting the group immediately.
He said if Iran threatened the US, "America will hold you fully accountable and, we won't be nice about it!"
In response, Hossein Salami, the top commander of Iran's Revolutionary Guards, said the Houthis made their own decisions.
"We warn our enemies that Iran will respond decisively and destructively if they carry out their threats," he told state media.
United Nations Secretary-General Antonio Guterres called for "utmost restraint and a cessation of all military activities" in Yemen and warned new escalation could "fuel cycles of retaliation that may further destabilise Yemen and the region, and pose grave risks to the already dire humanitarian situation in the country".
US Secretary of State Marco Rubio told CBS News' Face the Nation program: "There's no way the ... Houthis would have the ability to do this kind of thing unless they had support from Iran.
"And so this was a message to Iran: don't keep supporting them, because then you will also be responsible for what they are doing in attacking Navy ships and attacking global shipping."
Russian Foreign Minister Sergei Lavrov called Rubio to urge an "immediate cessation of the use of force and the importance for all sides to engage in political dialogue".
Five children and two women were among the 53 people killed in the US strikes, said Anees Alsbahi, spokesperson for the Houthi-run health ministry.
Another 98 people were injured.
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