"During boom times it's easy to run a country but try running when it's crumbling down," Fernandez said urging Argentines to continue to support her and pledging never to give up as her late husband taught her.
"Never let go, not even in the worst moments," she said. "Because it's in the worst moments when the true colors of a leader of a country comes out."
Buenos Aires Mayor Mauricio Macri, a member of the opposition and a fervent Fernandez critic, praised the big protest via Twitter. "People are being heard nationwide joined by a single flag," Macri said.
But polls suggest neither side has a firm grip on people's sympathies.
Fernandez easily won re-election just a year ago with 54 percent of the vote over a divided opposition but saw her approval rating fall to 31 percent in a nationwide survey in September by the firm Management & Fit. The survey of 2,259 people, which had an error margin of about two percentage points, also said 65 percent of respondents disapproved of her opponents' performance.
Crime is the biggest concern for many of her critics.
Argentine newspapers and television programs provide a daily diet of stories about increasingly bold home invasion robberies, in which armed bands tie up families until victims hand over the cash that many Argentines have kept in their homes since the government froze savings accounts and devalued the currency in 2002. The vast majority of the crimes are never solved, while the death toll is rising.
Inflation also upsets many, as the government's much-criticized index puts inflation at about 10 percent annually, or as little as a third of the estimates of private economists. As a result, real estate transactions have slowed to a standstill, given the difficulty of estimating the future value of contracts. And unions that won 25 percent pay hikes only a few months ago are threatening to strike again unless the government comes up with more.
The phrase "Cristina or nothing" was stenciled onto the sides of buildings surrounding the Plaza de Mayo, the iconic square facing the presidential pink palace.
Demonstrators at Plaza de Mayo held signs accusing the president of arrogance. While some featured a lengthy list of demands. Others simply said "basta" — enough.
"I agree with many of the things she's done but I don't agree with her method," said Barbara Torino, a psychologist. "The state has to do everything within the confines of the law or else it can easily lead to state terrorism."
Associated Press writers Michael Warren, Almudena Calatrava and Emily Schmall in Buenos Aires; Frances D'Emilio in Rome; Jorge Sainz in Madrid; and Luis Andres Henao in Santiago, Chile, contributed to this report.