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Before he began shooting on Wednesday, the attacker also broadcast an anti-Semitic manifesto online.
Several German media outlets said the man acted alone in the attacks, in which a woman was killed outside the synagogue and a man in the kebab shop. Police said they had detained one person.
Regional broadcaster MDR said the lives of two people who were seriously injured were no longer in danger.
The video circulating online showed a young man with a shaven head reciting a short statement in broken English to a camera while sitting in a parked car.
"I think the Holocaust never happened," he began, before adding "feminism is the cause of decline in birth rates in the West", mentioning mass immigration and concluding: "The root of all these problems is the Jew."
"We worked with urgency to remove this content and will permanently suspend any accounts found to be posting or reposting content of this abhorrent act," she said.
In the video, the man drove to the synagogue, found the gates shut, swore, and after failing to force the gates open, shot several rounds into a woman passer-by.
German magazines Spiegel and Focus Online reported that the suspect was a 27-year-old German.
Interior Minister Horst Seehofer said the shooting was anti-Semitic, adding: "According to the federal prosecutors' office, there are enough indications that it was possibly a right-wing extremist motive."
A spokeswoman for the Halle municipal government said one shooting took place in front of the synagogue on Humboldt street and its accompanying cemetery, while a second burst of gunfire targeted the kebab shop in the eastern German city.
Max Privorozki, Halle's Jewish community chairman, described how the gunman tried to shoot his way into the synagogue.
"We saw via the camera system at our synagogue that a heavily armed perpetrator with a steel helmet and a gun tried to shoot open our doors," he told the Stuttgarter Zeitung newspaper. "The man looked like he was from the special forces ... But our doors held."
"We barricaded the doors from inside and waited for the police," he said, adding that about 70 or 80 people were inside the synagogue celebrating Yom Kippur.
'CALM, LIKE A PROFESSIONAL'
German Chancellor Angela Merkel's government voiced outrage over the attack on Yom Kippur and urged tougher action against anti-Semitic violence.
"That on the Day of Atonement a synagogue was shot at hits us in the heart," Foreign Minister Heiko Maas wrote on Twitter. "We must all act against anti-Semitism in our country."
Anti-Semitism is a particularly sensitive issue in Germany, which during World War 2 was responsible for the genocide of 6 million Jews in the Nazi Holocaust. About 200,000 Jews live today in the country of around 83 million people.
Rifat Tekin, who worked at the kebab outlet, said he was making a kebab for two construction workers when a perpetrator threw an explosive at the restaurant before shooting.
"He was very calm, like a professional," Tekin told n-tv television. "He didn't say anything. He just kept coming and shooting ... I was hiding behind the salad counter."
Another witness, Conrad Roesler, said that when the attacker started firing at the restaurant, "I hid in the toilet."
Broadcasters showed images of the alleged perpetrator dressed in combat garb including a helmet.
In Berlin, politicians advised police to step up security at Jewish institutions in the capital.
Despite comprehensive de-Nazification in the post-war era, fears of resurgent anti-Semitic hatred have never completely gone away, whether from fringe, far-right neo-Nazis or more recently from Muslim immigrants.
Occasional past attacks have ranged from the scrawling of Nazi swastikas on gravestones to firebombings at synagogues and even several murders. In recent years, cases of assault or verbal abuse, in some cases directed against people wearing traditional Jewish skullcaps, have raised an outcry.