Clashes between protesters and police have left at least 93 people dead, 1,500 wounded and more than 500 behind bars.
The opposition coalition, the Democratic Unity Table (MUD), held a news conference to discuss Sunday's unofficial referendum in which almost 7.2 million Venezuelans rejected President Nicolas Maduro's government.
The opposition says that vote has been structured to pack the constitutional assembly with government supporters and allow Maduro to eliminate the few remaining checks on his power, creating a Cuba-style system dominated by his socialist party.
White House press secretary Sean Spicer says the USA congratulates the Venezuelan people "for the huge turnout in the referendum" sponsored by the opposition to reject President Nicolas Maduro's planned constitutional revamp.
Venezuelans living in Costa Rica take part in an unofficial referendum launched by the Venezuelan opposition in San Jose, Costa Rica, 16 July 2017.
"Today, Venezuela stood up with dignity to say freedom does not go backwards, democracy is not negotiated", said Julio Borges, who leads the opposition-controlled legislature after the referendum results were announced."We don't want a fraudulent Constituent Assembly imposed on us".
Colombia's former President Andres Pastrana, left, embraces Lilian Tintori, left after a news conference of worldwide observers of a symbolic referendum in Caracas, Venezuela, Sunday, July 16, 2017.
Sunday's reported 7.2 million participation compared with 7.7 million opposition votes in the 2015 legislative elections that it won by a landslide and 7.3 million votes for the opposition in a 2013 presidential poll narrowly won by Maduro.
Canada and Mexico were among the countries that issued statements Sunday evening lauding the opposition vote. Despite the violence, millions of people voted to reject the rewriting of the constitution and to call for elections.
In early July, armed pro-government militias supporting Venezuela's controversial President Nicolas Maduro stormed the country's Parliament, and assaulted several opposition deputies amid the ongoing anti-government protests in the South American country.
The referendum Sunday was also marred by violence.
"One particular thing that makes this act unlawful is that they never consulted with the people if they people want to change the constitution".
Venezuelan bishops have warned that President Maduro is seeking to establish a "socialist, Marxist and militaristic" state as tensions continue to rise in the Latin American country.
The rector of the Central University of Venezuela, Cecilia García Arocha, said 6,492,381 people voted inside Venezuela and another 693,789 at polling stations overseas.
Maduro and his officials have repeatedly said the opposition's poll is not a legitimate electoral procedure and called the results "meaningless".
On three questions at Sunday's event, opposition supporters voted overwhelmingly - by 98 percent - to reject the proposed new assembly, urge the military to defend the existing constitution, and support elections before Maduro's term ends, according to academics monitoring the vote for the opposition.
In this context, "suspending the process to start the Constituent Assembly would be an important gesture", added Mogherini, insisting on the "clear signal" sent on Sunday by the opposition in the consultation against the process.
The opposition however has blamed a "paramilitary" gang for the shooting, which prosecutors said they would investigate. The country's chief prosecutor has recently broken with the ruling party.