Those opposed to Maduro have long maintained that the election is fraudulent, not least because the opposition's most popular leaders - the ones with the best chance of unseating the president - were barred from running.
"The process undoubtedly lacks legitimacy and as such we do not recognize it", said Falcon, a 56-year-oldformer state governor, looking downcast.
"For us, there were no elections".
He said Mr Maduro must allow humanitarian aid into Venezuela and that the U.S. will not "sit idly by".
Venezuela's socialist President Nicolas Maduro has faced worldwide condemnation after his re-election in a vote foes denounced as a farce.
"We won again! We triumphed again!" About 5.8 million votes went to Maduro and Falcon came in second with 1.8 million votes, according to the electoral board, with 92.6% of the votes tallied.
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Maduro, the political heir to the late leftist firebrand Hugo Chavez, has presided over an implosion of the once wealthy oil producer's economy since taking office in 2013.
Hyperinflation, food and medicine shortages, rising crime and broken water, power and transport networks have sparked violent unrest, and left Maduro with a 75 per cent disapproval rating.
Venezuela ejected the Canadian charge d'affairs - who is the top person at an embassy in the absence of an ambassador - in December 2017 over the federal government's criticism of the human rights situation there.
A woman waiting outside a polling station in a Caracas slum, La Vega, said she worked for a government agency and feared losing her job if she did not vote and report afterward at the Red Spot.
Sen. Marco Rubio, R-Florida, called on those leaders to collectively reject the Venezuela vote and praised the administration for imposing additional sanctions against the Maduro regime.
He added that he would be open to dialogue with the "empire" of the United States.
"Sham elections change nothing".
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Government critics also say other voters were pressured into voting for Maduro.
The government has set up so-called red point zones near polling stations so Venezuelans can scan their state-issued "fatherland cards" used to receive benefits including food boxes and money transfers.
Voting centers across Venezuela appeared largely empty for the election despite assurances from government officials that millions had turned out to vote by midmorning. Hundreds of thousands of Venezuelans have fled the nation and starvation is rampant.
A man chooses his candidate during presidential elections in Caracas, Venezuela, May 20, 2018.
Falcon said fresh elections could be held in November or December, when they are traditionally contested, but they were moved up this year by the country's all-powerful and pro-government Constituent Assembly, catching the divided and weakened opposition off-guard.
As voting took place on Sunday, a senior State Department official warned that the USA might press ahead on threats of imposing crippling oil sanctions, according to the AP.
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