"Attacks against health facilities in Syria by the regime or its supporters are unacceptable and must stop immediately," said French Foreign Minister Jean-Marc Ayrault.
The United States, which like the UN did not identify who carried out the air strikes, said two civilian hospitals were hit in northern Syria: one run run by medical charity Medecins Sans Frontieres (MSF) and another in rebel-held Azaz city.
The region around Syria's second city of Aleppo has been the target of a major offensive by Syrian government troops, backed by Russian warplanes, which has sent tens of thousands fleeing to the Turkish border.
"That the Assad regime and its supporters would continue these attacks... casts doubt on Russia's willingness and/or ability to help bring to a stop the continued brutality of the Assad regime against its own people," the State Department said.
MSF confirmed a hospital supported by the charity was hit in Idlib, northwest Syria, and said seven people were killed and at least eight were missing, presumed dead.
But Syria's ambassador to Moscow, Riad Haddad, said the hospital had been targeted by a US raid.
"American warplanes destroyed it. Russian warplanes had nothing to do with any of it -- the information that has been gathered will completely back that up," he told Russian state television channel Rossiya 24.
- Kurds make fresh gains -
The heated words cast fresh doubt on a days-old deal to enforce a nationwide ceasefire later this week, with Syria's President Bashar al-Assad saying it would be "difficult" to implement.
"They are saying they want a ceasefire in a week. Who is capable of gathering all the conditions and requirements in a week? No one," Assad said in televised remarks.
Seeking to keep alive the deal for a "cessation of hostilities" in Syria clinched in Munich, Germany, last week, the UN's envoy for Syria, Staffan de Mistura, arrived in Damascus on Monday.
Kurdish fighters in northern Syria meanwhile captured a key town, despite Turkey resuming shelling in several parts of Aleppo, alarmed by their recent advances against mostly Islamist rebels.
Ankara accuses the Kurdish forces of ties to the outlawed Kurdistan Workers Party (PKK) that has waged a decades-long insurgency against Turkey.
Britain-based monitor the Syrian Observatory for Human Rights said at least two children died in Monday's shelling and an AFP journalist on the border said Turkish howitzers fired for around 20 minutes from the Akcabaglar region.
Nonetheless, the coalition of Kurdish and Arab fighters known as the Syrian Democratic Forces (SDF) captured the town of Tal Rifaat from mostly Islamist rebels, the Observatory said.
The SDF had already seized the nearby Minnigh airbase and severed the road between Tal Rifaat, which lies 20 kilometres (12 miles) from the border, and rebel-held Azaz.
Turkey fears the Kurds will be able to create a contiguous Kurdish territory just across the border in northern Syria.
Turkish Prime Minister Ahmet Davutoglu warned earlier Monday that Ankara would not let Azaz fall to the SDF, adding that "the necessary intervention will be made".
- EU joins condemnation -
Moscow says its military intervention has targeted the Islamic State jihadist group and other "terrorists", but activists say Russia's raids have caused disproportionately high civilian casualties.
Russia has been accused of hitting several health facilities since its Syrian aerial campaign began September 30.
EU foreign policy chief Federica Mogherini branded the Idlib attack "unacceptable" and urged "all parties (to respect) basic principles of humanitarian law".
The Observatory also reported 10 civilians, including three children, were killed in suspected Russian strikes in and nearby Azaz that hit near a hospital, it said.
Russia's air strikes have allowed government forces to press a major operation that has virtually encircled rebels in eastern Aleppo city, as well as pushing them from much of the region to the north.
They have angered Turkey, with Davutoglu on Monday warning: "If Russia continues behaving like a terrorist organisation and forcing civilians to flee, we will deliver an extremely decisive response."
Moscow called Turkey's shelling in Syria "provocative" and said it supported raising the issue at the UN Security Council.
A US State Department spokesperson urged Turkey and Russia to avoid any further escalation.
"It is important that the Russians and Turks speak directly, and take measures to prevent escalation," the spokesperson told AFP.
More than 260,000 people have been killed since Syria's conflict began with anti-government protests in March 2011.