The attack, which authorities said injured women in labour and left children in the wreckage, underscored US warnings that the biggest assault on a European state since 1945 could become increasingly attritional after Russia's early failures.
Russia's invasion has entered its third week with none of its key objectives reached despite thousands of people killed, more than two million made refugees, and thousands forced to cower in besieged cities under relentless bombardment.
Moscow's stated objectives of crushing the Ukrainian military and ousting the pro-Western elected government of President Volodymyr Zelenskiy remained out of reach, with Zelenskiy unshaken and lethal Western military aid pouring across the Polish and Romanian borders.
Western-led sanctions designed to cut the Russian economy and government from international financial markets were beginning to bite, with the Russian sharemarket and rouble plunging and ordinary Russians rushing to hoard cash.
The White House condemned the hospital bombing in Mariupol on Wednesday as a "barbaric use of military force to go after innocent civilians".
Russian had earlier pledged to halt firing so at least some trapped civilians could escape the port city, where hundreds of thousands have been sheltering without water or power for more than a week. Both sides blamed the other for the failure of the evacuation.
"What kind of country is this, the Russian Federation, which is afraid of hospitals, is afraid of maternity hospitals, and destroys them?" Zelenskiy said in a televised address late on Wednesday.
Zelenskiy repeated his call for the West to tighten sanctions on Russia "so that they sit down at the negotiating table and end this brutal war". The bombing of the children's hospital, he said, was "proof that a genocide of Ukrainians is taking place".
Kremlin spokesman Dmitry Peskov, asked by Reuters for comment, said: "Russian forces do not fire on civilian targets." Russia calls its incursion a "special operation" to disarm its neighbour and dislodge leaders it calls "neo-Nazis."
The UN Human Rights body said it was verifying the number of casualties at Mariupol. The incident "adds to our deep concerns about indiscriminate use of weapons in populated areas," it added through a spokesperson.
Among more than 2 million total refugees from Ukraine, the United Nations Children's Fund (UNICEF) said on Wednesday that more than 1 million children have fled the country since the invasion started on February 24. At least 37 had been killed and 50 injured, it said.
Russian Foreign Minister Sergei Lavrov is due to hold talks on Thursday with his Ukrainian counterpart Dmytro Kuleba, the first meeting between the two since the invasion.
Ukraine is seeking a ceasefire, liberation of its territories and to resolve all humanitarian issues, Kuleba said, adding: "Frankly ... my expectations of the talks are low."
Moscow demands that Kyiv take a neutral position and drop aspirations of joining the NATO alliance.
Zelenskiy told VICE in an interview on Wednesday that he was confident Putin would at some stage agree to talks. "I think he will. I think he sees that we are strong. He will. We need some time," he said.
Russia has been hit by Western sanctions and the withdrawals of foreign firms. Nestle, cigarette maker Philip Morris and Sony on Wednesday joined the list of multinationals stepping back from the country.
The World Bank's chief economist said Moscow was edging close to defaulting on its debt. The Kremlin is taking measures to shore up the economy and planned to respond to a US ban on its oil and energy exports as the rouble dropped to record lows.
The US House of Representatives on Wednesday voted to rush $US13.6 billion in aid to Ukraine, sending the legislation to the Senate.
