However, in response to these voices, the Venezuelan bishops have decried the consultation "legitimate" as it represents the Venezuelan people's constitutionally-endowed sovereignty.
Meanwhile, the opposition praised the vote as proof that Venezuelans demand democratic change.
The figure obtained by the opposition, however, is lower than the 7,587,579 votes secured by Maduro in the presidential elections of 2013, when he beat opposition leader Henrique Capriles by less than two percentage points.
Some supporters said they were disappointed.
I'm not sure there is any going backwards from here.
The opposition also said the country's National Assembly, which it controls, would name new members to the government-dominated Supreme Court, setting up a showdown with President Nicolas Maduro, whose party controls almost all other state institutions.
Maduro has called for the creation of a "constituent assembly", a Soviet-like body that will seek to modify the country's constitution. Venezuela's opposition called for a massive turnout Sunday in a symbolic rejection of Presid.
Fox said he left Venezuela as planned Monday and hadn't been personally notified of any action by the Venezuelan government.
"I don't think words can explain - you just stand here and look around", said Spring Hill resident Stephanie Labrecque, referring to the large line of people waiting to vote in the parking lot at Are Pitas restaurant on University Square drive just north of Fowler Avenue.
Venezuelan Foreign Minister Samuel Moncada said former Mexican president Vicente Fox, who was in Venezuela for the vote, had been declared persona non grata and banned him from the country for conspiring to instigate violence and foreign intervention.
The president of Venezuela plans to hold a vote to elect a special assembly to rewrite the constitution on July 30. Moncada did not provide evidence to back his claims.
The chief prosecutor's office said Xiomara Soledad Scott, a nurse, had been killed and four others wounded.
Video posted to social media showed huge crowds outside the church before running in panic as men on motorbikes zoomed past and shots rang out.
Speaking on state television shortly after the polls closed, Maduro failed to make any mention of the shooting.
"They (MUD) have a quiet consultation today (.) I tell them not to go insane, we make a call for them to return to peace, to sit down and talk, to begin a new cycle of dialogue for peace", said Maduro during a telephone interview with the state-owned Venezolana de Televisión. "Let's start a new round of talks, of dialogue for peace".
In smaller numbers in many parts of the capital, government supporters also went to polling stations on Sunday in a rehearsal for the July 30 vote. "Today we're following his legacy, with President Nicolas Maduro ..."
The opposition is calling on the government to call off the July 30 election of members of a constitutional assembly that would remake the country's political system.
The opposition has criticized Maduro's Constituent Assembly as an attempt to establish a dictatorship in which the National Assembly's powers would be further weakened.
"The ruling Socialist Party can not win a free and fair election of any kind, and the Constituent Assembly is created to resolve their collective electability problem by tilting the electoral playing field", Eurasia consultancy said. "There's no separation of powers, no freedom of expression".
The National Assembly - also controlled by the opposition - says it will name new members of the country's supreme court, a move certain to be blocked by Maduro's administration.
In asking the question in the unofficial referendum what role the military should play in the current crisis, the opposition appears to be trying to apply pressure on parts of the military to break with Maduro. Do they reject the constitutional assembly? The people expressed themselves and told the Executive what they want: they want peace.
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.
Venezuela has been plagued by daily outbursts of anger over food shortages and an economic crisis characterised by rampant inflation. More than 500 people have been detained.