The Western-backed alliance fighting the Iranian-aligned Houthi group in Yemen said in a statement that the air strikes targeted missile launchers used to attack the southern Saudi city of Jizan on Wednesday, killing a Yemeni civilian there.
It accused the Houthis of using children as human shields and said the strikes were carried out in accordance with international humanitarian law.
Houthi spokesman Mohammed Abdul-Salam said the coalition showed "clear disregard for civilian life" as the attack had targeted a crowded public place in the city.
The ICRC said a strike hit the bus driving children in Dahyan market, in northern Saada.
"Our shops were open and shoppers were walking around as usual. All of those who died were residents, children and shop owners," witness Moussa Abdullah, who was being treated in hospital for wounds, told Reuters.
The ICRC said on its Twitter account that its medical team at the ICRC-supported hospital in Saada had received the bodies of 29 children, all under 15 years old. The hospital also received 48 wounded people, among them 30 children.
That was the toll at just one hospital.
Abdul-Ghani Sareeh, from Saada health department, told Reuters: "A bus carrying children was targeted today while they were coming from summer school resulting in 43 martyrs and almost 63 wounded."
White plastic body bags filled the floor of a room in the ICRC-supported hospital.
Injured children, bloodied, bandaged and screaming, lay on stretchers as doctors treated them, friends and relatives having carried some of them in their arms to be treated.
"Scores killed, even more injured, most under the age of 10," Johannes Bruwer, head of the delegation for the ICRC in Yemen, said earlier in a Twitter post.
It was unclear how many of the dead in total were children and how many air strikes were carried out in the area, in northern Yemen, near the border with Saudi Arabia.
Saudi Arabia and Sunni Muslim allies intervened in Yemen's war in 2015 against the Houthis, who control the most populous areas of Yemen, including the capital Sanaa, and drove the internationally recognized government into exile in 2014.
The United States and other Western powers provide arms and intelligence to the alliance, and human rights groups have criticized them over coalition air strikes that have killed hundreds of civilians at hospitals, schools and markets.
"Grotesque, shameful, indignant. Blatant disregard for rules of war when bus carrying innocent school children is fair game for attack," Jan Egeland, head of the Norwegian Refugee Council, said in a Twitter post.
The World Health Organisation (WHO) representative in Yemen, Nevio Zagaria, said it has deployed emergency supplies. "I am extremely saddened by what happened in Saada ... The attack on civilians is not acceptable."
The alliance says it does not intentionally target civilians and has set up a committee to probe alleged mass casualty air strikes, which has mostly cleared the coalition of any blame.
"Today's attack in Saada was a legitimate military operation ... and was carried out in accordance with international humanitarian law," the coalition said in the Arabic-language statement carried by SPA.
"Targeting Saudis and residents in Saudi is a red line," coalition spokesman Turki al-Malki later told Al Arabiya TV.
Fragments from the Houthi missile launched at Jizan Industrial City had killed one Yemeni civilian and wounded 11, Saudi state media said earlier on Thursday.
The Houthis have launched a series of missile strikes on the kingdom, including Riyadh, over the past year.
Saada, the main stronghold of the Houthis, has mainly come under air strikes from the coalition as the mountainous province makes battles hard for pro-government ground troops.
The Yemen war has killed more than 10,000 people, displaced more than 2 million and driven the country to the verge of famine, according to the United Nations.
(Reporting by Dubai Newsroom; Writing by Aziz El Yaakoubi; Additional reporting by Stephanie Nebahay in Geneva; Eding by Alison Williams)