Trump said the attacker Sayfullo Habibullaevic Saipov, who ploughed a pickup truck down a crowded bike path near the World Trade Centre killing eight people and injuring 11 others in an ISIS-inspired plot, was allowed to enter the US on a State Department programme known as the "Diversity Lottery Programme".
"The terrorist came into our country through what is called the 'Diversity Visa Lottery Programme', a Chuck Schumer beauty. I want merit based," Trump said in a tweet.
Saipov, a 29-year-old Uzbek immigrant, had come to the US legally in 2010, was shot at by New York Police officials yesterday afternoon and is being treated for injuries at a New York hospital.
Local media reports said that he was being subject to preliminary interrogation by investigating officials.
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The FBI and New York Police Department are jointly investigating the terrorist attack.
New York Governor Andrew Cuomo told a news channel that the sole attacker was radicalised domestically.
"The evidence shows - and again, it's only several hours, and the investigation is ongoing - but that after he came to the United States is when he started to become informed about ISIS and radical Islamic tactics," Cuomo told CNN.
Trump has ordered the Homeland Security to step up extreme vetting programme.
In a series of tweets, the President lashed out at Senate Minority Leader Chuck Schumer for opposing merit-based immigration system.
"Senator Chuck Schumer helping to import Europe's problems.. We will stop this craziness!" Trump tweeted.
"We are fighting hard for Merit Based immigration, no more Democrat Lottery Systems. We must get MUCH tougher (and smarter)," he said.
"President Trump, instead of politicising and dividing America, which he always seems to do at times of national tragedy, should be focusing on the real solution anti- terrorism funding which he proposed cutting in his most recent budget," Schumer said.
"I'm calling on the president to immediately rescind his proposed cuts to this vital antiterrorism funding," he said.
"This was an act of terror, and a particularly cowardly act of terror aimed at innocent civilians, aimed at people going about their lives who had no idea what was about to hit them," New York Mayor Bill de Blasio said.
The most recent violence from terrorism there came in September 2016, when a man set off shrapnel-packed explosives in the Chelsea neighbourhood of Manhattan. Nobody was killed, but 30 people were injured in the incident.