"On behalf of the United States, I'd like to extend my really honest condolences to a friend of mine, President [Emmanuel] Macron of France, where they just yesterday had a vicious, vicious Islamic terrorist attack - beheading an innocent teacher near Paris", he said.
Four relatives of the attacker, including a minor, were detained in the immediate aftermath of the attack in the middle-class suburb of Conflans-Sainte-Honorine.
The father of one schoolgirl had launched an online call for "mobilization" against the teacher and had sought his dismissal from the school.
His half-sister joined the Islamic State group in Syria in 2014, Ricard said.
The source said the killing was carried out by an 18-year-old Chechen, who was then shot and killed by police near the scene in Conflans-Sainte-Honorine, northwest of Paris. Police have yet to provide further details of the incident, which they have labeled a terrorist attack.
A 10th person was placed in custody later on Saturday.
A text claiming responsibility for the attack and a photograph of the victim were found on the suspect's phone, he said, adding that the suspect had been seen at the school asking students about the teacher, and the headmaster had also received several threatening phone calls.
Ricard said Paty's murder illustrated "the very high level terrorist threat" France still faces.
Armed police killed the attacker minutes later near the College du Bois d'Aulne school in the suburb. Others held signs saying, "I am a teacher".
Thousands of people rallied in central Paris Sunday in a defiant show of solidarity with a teacher beheaded for showing pupils cartoons of the Prophet Mohammed.
Martial, a 16-year-old pupil, said Paty had loved his job: "He really wanted to teach us things".
Nine arrested in France over beheading of teacher as Macron denounces 'Islamist terrorist attack'
Ricard, the prosecutor, did not say if the attacker had any links to the school, pupils or parents, or had acted independently in response to the online campaign.
The killing was described by French President Emmanuel Macron on Friday as a "terrorist attack".
"One of our citizens was murdered today because he was teaching, he was teaching students about freedom of expression, the freedom to believe and not believe".
Prime Minister Jean Castex said the attack bore the hallmarks of Islamist terrorism. "They won't win. We will act. Firmly, And quickly. You can count on my determination".
Officials and Muslim leaders throughout France expressed outrage at the slaying.
Part of that population is made up of Chechens. In the 1990s, two wars in Chechnya, a predominantly Muslim Russian republic in the North Caucasus, triggered a wave of emigration and many fled to western Europe. The outlet published caricatures of the Prophet Muhammad, unleashing divisions that are still casting a pall over French society.
Last month, a young man from Pakistan was arrested after stabbing two people with a meat cleaver outside the former offices of the satirical newspaper Charlie Hebdo.
The weekly was the target of a deadly newsroom attack in 2015, and it republished caricatures of the prophet this month to underscore the right to freedom of information as a trial opened linked to that attack.
Friday's attack was the second of its kind since a trial started last month over the Charlie Hebdo killings. Prior to the said attack, the magazine's offices were struck by a gasoline bomb in 2011.
"We're here to defend the Republic, the values of the Republic: liberty, equality, fraternity and secularism". Adela Suliman reported from London.