Nearly 7.2 million Venezuelans took part in the non-binding referendum organized by the country's main opposition parties, with the overwhelming majority of voters coming out against Maduro's plan.
Venezuela is in the grip of social unrest, which has paralyzed the country amid violent protests seeking government reform.
The government has condemned the referendum as illegal, and has instead called for a July 30 vote to elect a special assembly to rewrite the 1999 constitution.
The referendum asked voters three yes-no questions. More than 98% of voters chose to reject the proposed constitutional assembly; request the military defend the existing constitution; and support fresh elections before Maduro's term ends in 2019.
Cecilia Garcia Arocha, rector of Central University of Venezuela, announced the results at the referendum headquarters in Caracas, pointing out that the number of polling stations, at 2,030, was much lower than in regular elections.
Sunday's turnout represents about 37% of Venezuela's total electorate, according to CNN calculations based on 2015 National Electoral Committee figures.
In 2013, Maduro was elected with just over 7.5 million votes, and Henrique Capriles came in second with 7.3 million votes.
In response to Sunday's results, Venezuela's National Assembly Vice President Freddy Guevara called for a 24-hour nationwide strike to take place Thursday. The strike is an effort to increase pressure on Maduro.
The "civil strike this Thursday cannot come only from businessmen. The country belongs to everyone and everyone needs to guarantee to paralyze Venezuela," Guevara wrote on Twitter.
At least one person was killed in the western Caracas neighborhood of Catia and several more injured as shots were fired at a polling station.
The attorney general's office said authorities were investigating the "irregular circumstances" of the woman's death, caused by armed attackers on motorcycles.
Separately, the mayor of a neighboring municipality and opposition spokesperson Carlos Ocariz told CNN that pro-government armed militias known as colectivos fired shots at the referendum polling station; he said that two people were killed.
Marialbert Barrios, congresswoman for the region of Catia, said the colectivos tried "to stop the right of a people to express their will for a better country." When they opened fire, Barrios said, most of the voters took cover, "but at least one middle-aged lady lost her life."
The attack took place just meters away from a school where Maduro was expected to speak. "This shows the government fears us," Barrios told CNN. "This shows that Maduro is afraid of a people who votes in freedom."
The government intimidates and restricts the media in Venezuela, taking CNN en Español off the air. The government tightly controls visas for foreign journalists including CNN, arresting those who report from inside the country without proper permits.
At the same time, Maduro's supporters also cast a ballot -- with the government staging a rehearsal of the July 30 constitutional elections.
Competing lines could be seen throughout Caracas, with opposition and government approved polling stations only meters apart.
In the central neighborhood of La Candelaria, voter Lysett Alvarez said the proposed constitutional assembly would be fraudulent and illegal, and she was firmly against it. "I am here with all my neighbors, everyone in the area came down to defend freedom, to demand for more medicine and to ask for a Venezuela in peace. What we want is peace!"
"This government must leave, here and now. This is our last chance," added voter Carlos Macia, who said he was fighting for his own future together with many young Venezuelans.
As the results came in, Foreign Minister Samuel Moncada took to Twitter to declare that former Mexican President Vicente Fox had been banned from the country for promoting violence and supporting foreign intervention.
Fox had traveled to Venezuela on Saturday
as part of a contingent of former Latin American presidents to show their support for the referendum.
The opposition also set up 667 polling stations in 100 different countries for expats who wanted to make their voices heard.