Maduro on Wednesday accused the international firm behind the claim of bowing to US pressure to cast doubt over what he called a "transparent vote".
Smartmatic, the technology firm hired to handle Sunday's vote, said the official figure of 8.1 million turnout was "tampered with", prompting Attorney General Luisa Ortega to open an investigation.
The assembly, tasked with rewriting Venezuela's constitution, was originally due to start work Thursday, but Maduro postponed the launch to Friday in the face of opposition plans for massive protests.
In his first meeting with assembly delegates on Wednesday night, Maduro not only stood by the official count, but proclaimed that an additional two million people would have voted if they had not been blocked by anti-government protesters.
Maduro called the vote in May after weeks of protests fuelled by widespread anger over food shortages, triple-digit inflation and high crime - unrest that continues and has caused at least 125 deaths.
"That stupid guy, the president of Smartmatic, pressured to the neck by the gringos and the Brits, said there were 7.5 million," Maduro said in televised remarks. "I think there were 10 million Venezuelans who went out."
Maduro provided no evidence to support his claim, but his remarks were received with resounding applause from about 500 people elected to the assembly.
Antonio Mugica, chief executive of Smartmatic, said there was glaring evidence of vote-tampering.
"We estimate the difference between the actual participation and the one announced by authorities is at least one million votes," he said.
Smartmatic has provided voting technology in Venezuela since 2004.
Ortega, the attorney general, said the company's assessment was just "one more element of the fraudulent, illegal and unconstitutional process" initiated by the socialist ruler.
"We are facing an unprecedented, serious incident that represents a crime," Ortega told CNN.
Tibisay Lucena, the head of the National Electoral Council, dismissed Smartmatic's claim, saying it was an "opinion" of a company that played only a secondary role in the election and had no access to complete data.
"A company located outside the country does not guarantee the transparency and credibility of the Venezuelan electoral system," Lucena said.
Sunday's vote drew international condemnation.
The United States imposed direct sanctions on Maduro, calling him a "dictator," while the European Union joined the US, Mexico, Colombia and Argentina in saying it would not recognise the new assembly.
But oil workers, whom Maduro considers a bedrock of support, rallied across Venezuela on Wednesday to denounce sanctions on the leftist president.
"We are here to show our rejection of the intervention of the United States," one demonstrator said during a televised rally, calling the sanctions "a political show with harmful economic consequences for the people of Venezuela".