Turki al-Maliki said the kingdom's air defense had intercepted a ballistic missile fired by the Houthi rebels at a densely populated civilian area in Jizan, according to the Saudi Press Agency.
Johannes Bruwer, the head of the International Committee of the Red Cross (ICRC) delegation to Yemen, said in a tweet that most of those killed by the airstrike were children less than 10 years of age.
The ICRC said in its tweet: "Under global humanitarian law, civilians must be protected during conflict".
The Huthi' Al-Masirah TV reported that 39 people had been killed and 51 wounded, "mostly children".
US State Department spokeswoman Heather Nauert said the United States did not have the "full details of what happened on the ground" but said "we're concerned about these reports".
The missile was sacked from Amran province in northern Yemen on Wednesday night, the coalition said in a statement published by the Saudi Press Agency.
Last week, Yemeni medical officials said the coalition conducted airstrikes in the rebel-held port city of Hodeida, killing at least 28 people and wounding 70.
The attack, the spokesperson said, was retaliation for a missile fired into Saudi Arabia Wednesday, killing one and injuring 11 more.
The Red Cross, however, confirmed that "scores" of children were killed, saying that most of the victims were under the age of 10.
Col. Al-Maliki stressed: "The Coalition will take all necessary measures against the terrorist, criminal acts of the Iranian-Houthi militias, such as recruiting child soldiers, pushing them into battlefields and using them as shields for their terrorist acts". Most of the images of the children hit by this airstrike are so gruesome the broadcasting rules don't allow us to show them - but a warning, the report still contains distressing images.
A child lies in hospital after the school bus he was on was allegedly attacked by the Saudi-led coalition. The paramedics present at sites reportedly said that they had found and treated 40 people with broken bones, cuts and bruises.
But on social media, critics noted that other coverage of the air strike in the U.S. and Britain-which also supports the coalition-failed to acknowledge the countries' involvement in the war.
Col Al Maliki said the coalition was not intentionally targeting civilians, saying the forces used "a high standard measure for targeting, ... civilian casualties are a loss to the coalition and we can not accept high civilian casualties in Yemen".
Wednesday's attack brings the tally to 167 rebel missiles launched since 2015, according to the coalition, which that year joined the Yemeni government's fight against Houthi rebels.
Impoverished Yemen, on the southern tip of the Arabian Peninsula, is now in the world's worst humanitarian crisis, with more than 22.2 million people in need of assistance.
A USA military spokeswoman said United States forces were not involved in Thursday's air strike.