National Assembly President Juan Guaido made the statement to an energized crowd blocking a busy Caracas street a day after Maduro's inauguration to a second term.
"Guaido for president!" the crowd chanted. "Out with Maduro!"
"The constitution gives me the legitimacy to carry out the charge of the presidency over the country to call elections," Guaido said. "But I need backing from the citizens to make it a reality."
The head of the Organization of American States, Secretary-General Luis Almagro, wasn't waiting. He sent out a tweet recognizing Guaido as Venezuela's interim president. "You have our support," Almagro said in a tweet.
Reiterating the U.S. position that the May election that gave Maduro a second term was "not free, fair or credible," Bolton said that "we support the courageous decision" of Guaido's declaration "that Maduro does not legitimately hold the country's presidency."
Guaido asked Venezuelans to mass in a nationwide demonstration on Jan. 23, a historically important date for Venezuelans — the day when a mass uprising overthrew dictator Marcos Perez Jimenez in 1958.
The constitution assigns the presidency to the head of the National Assembly if Maduro is illegitimate.
But the overall military so far has remained firmly behind Maduro, despite some reports of small-scale attempts at revolt.
A once wealthy oil nation, Venezuela is gripped by growing crisis of relentless inflation, food shortages and mass migration.
The announcement is a daring challenge to the socialist leader, who has rejected criticism of his re-election and whose government has imprisoned many leading critics.
Maduro accuses the United States and local foes of plotting a coup.
Seventeen Latin American countries, the United States and Canada denounced Maduro's government as illegitimate in a measure adopted Thursday at the Organization of American States in Washington.
In May, Maduro declared victory following an election that his political opponents and many foreign nations consider illegitimate, in part because popular opponents were banned from running and the largest anti-government parties boycotted the race.
Friday's demonstration was the largest showing of anti-government supporters in more than a year, but fell far short of the thousands that took to the streets over four months in 2017, leading to clashes in which more than 120 died.
Guaido, 35, made the announcement less than a week after being selected to lead Venezuela's National Assembly, vowing to press for transition of power.
Guaido has won some international support, speaking by phone this week with U.S. Secretary of State Mike Pompeo.
"We are not victims here. We are survivors and we are going to survive this," Guaido said. We are here to talk about the route, because there are no magical solutions."