Venezuela is in crisis, and more than 100 people have died in political clashes since April.
More than 7m people cast ballots in Sunday's vote, according to a group of academics monitoring the referendum, which has no legal status.
The rector of the Central University of Venezuela, Cecilia García Arocha, said 6,492,381 people voted within the country and another 693, 789 at polling stations abroad.
The turnout is slightly less than the 7.7m people who voted for opposition candidates at the 2015 parliamentary elections. There are 19.5m registered voters in the country.
Voting yes or no to three questions, 98% rejected the new assembly proposed by President Nicolas Maduro and backed a call for elections before his term of office ends in 2019.
They also voted for the armed forces to defend the current constitution.
The opposition blamed a "paramilitary" gang for the shooting incident. Video from the scene showed people rushing away from the gunshots, as many fled to a church.
Opposition spokesman Carlos Ocariz said of the shooting: "We lament this very much, with great pain."
Prosecutors said they would investigate the incident, and the woman was named as Xiomara Soledad Scott.
She died minutes after reaching hospital.
Separately, journalist Luis Olavarrieta was kidnapped, robbed and beaten by a group of people. He managed to escape and images emerged of him receiving medical attention.
An official vote will be held on 30 July for a new assembly, which would have the power to rewrite the constitution and to dissolve state institutions.
But critics say the new assembly could herald dictatorship.
Opposition politicians organised Sunday's unofficial poll, held in improvised polling stations at theatres, sports grounds and roundabouts within Venezuela and in more than 100 countries around the world.
While the vote was only symbolic, BBC South America correspondent Katy Watson said the opposition hoped that a high turnout would heap pressure on the government.
Queues started to form early and there was a festive atmosphere in most places.
But President Nicolas Maduro described Sunday's vote as "meaningless".
"They have convened an internal consultation with the opposition parties, with their own mechanisms, without electoral rulebooks, without prior verification, without further verification. As if they are autonomous and decide on their own," he said.
Catia, where the shooting happened, is a poorer part of the Venezuelan capital where support for the socialist government, headed first by Hugo Chavez and since his death by Mr Maduro, has historically been high.
Mr Maduro argues that the constituent assembly is the only way to help Venezuela out of its economic and political crisis.
He has said that a new constitution would "neutralise" the opposition and defeat "coup-plotters" and thereby promote peace in Venezuela.
Opposition leaders fear that the process of setting up a new constituent assembly and rewriting the constitution would almost certainly delay this year's regional elections and next year's presidential election.
They also fear that the constituent assembly would further weaken the National Assembly, Venezuela's opposition-controlled legislative body.