The knife-wielding suspect, later identified by prosecutors as a 37-year-old Algerian-born man, was arrested at the site of Saturday's attack in the eastern city of Mulhouse.
Local prosecutor Nicolas Heitz said the suspect, who he did not name, was registered on France's terrorist watchlist.
A further three people were in custody in connection with the case Sunday, the PNAT prosecutors unit said, without giving details.
Speaking at the police station, Interior Minister Bruno Retailleau said the man had "a schizophrenic profile" and his act had "a psychiatric dimension".
Retailleau said France had repeatedly attempted to expel him from the country, but Algeria refused to cooperate.
The rampage occurred around 4 pm (1500 GMT) near a busy market in Mulhouse, a city of around 110,000 people near the German border. At the time, demonstrators were rallying in support of the Democratic Republic of Congo.
A 69-year-old Portuguese man was fatally wounded while parking attendants and police were also hurt.
Two officers were seriously wounded, with one sustaining an injury to a carotid artery, and the other to the upper body, prosecutor Heitz told AFP, adding that the latter officer was able to leave hospital.
Three other officers suffered minor injuries, prosecutors said.
During the attack, the suspect was heard shouting "Allahu Akbar" (God is greatest), according to the national anti-terror prosecutors unit (PNAT).
Witnesses also told AFP they heard the suspect shouting the words several times.
President Macron later said there was "no doubt" that the incident was "a terrorist act", specifically "an Islamist terrorist act".
The government was determined to continue doing "everything to eradicate terrorism on our soil", he added.
Speaking during a visit to France's agriculture fair, Macron offered condolences to the family of the victim and said the "solidarity of the nation" was behind them.
PNAT said it was investigating the attack for murder and attempted murder "in connection with a terrorist enterprise".
Failed expulsions
The terrorist watchlist, called FSPRT, compiles data from various authorities on individuals with the aim of preventing "terrorist" radicalisation.
It was launched in 2015 following deadly attacks on satirical magazine Charlie Hebdo's offices and on a Jewish supermarket.
Interior Minister Retailleau told French broadcaster TF1 that France had tried to expel him 10 times, with Algeria refusing each time to accept him.
"Once again, it is Islamist terrorism that has struck," he said. And, once again, he added, problems of migration were "at the origin of this terrorist act".
There was no immediate comment from Algeria's presidency or foreign ministry.
As night fell, several members of the forensic police were still working under the glow of a spotlight outside the covered market in Mulhouse. The perimeter was guarded by military personnel.
Prime Minister Francois Bayrou said that "fanaticism has struck again, and we are in mourning".
Mulhouse Mayor Michele Lutz wrote on Facebook that "horror has just seized our city".
France has recently experienced a string of stabbings deemed acts of terror.
In January, a 32-year-old knife-wielding man wounded a person in a supermarket in Apt, in the south of France. He was charged and jailed for attempted murder in connection with a terrorist undertaking.
In December 2023, a man suspected of stabbing a German tourist to death near the Eiffel Tower was charged with carrying out a terror attack.
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