In a brief and quiet ceremony inside the green-domed Congress Building, Lt. Gen. Roberto A. Viola was sworn in today as the 37th president of Argentina.

Around the edges of the Plaza de Mayo, the palm-fringed square that fronts the pink presidential palace, a small crowd waited this morning under rain clouds for the first sign of the presidential limousine. The looked to be outnumbered by the uniformed police who strolled the plaza, lined the empty avenue and gazed down from nearby roofs.

At 10:15, Army Commander Leopoldo Galtieri draped the blue-and-white sash of the presidency over Viola's white-jacketed chest. Then he handed Viola the presidential scepter and shook his hand. With that, Viola became the second Army general to rule Argentina since the military took power in a 1976 coup.

The general who led that coup, retiring president Jorge R. Videla, embraced Viola moments after the presidency was officially handed over. Like Videla, Viola was selected as president by the three-man military junta that officially is to share power with the president until the restoration of a "republican and representative democracy" some time in the future.

Viola spoke of that democracy in comments to reporters at the presidential palace where the new Cabinet was sworn in. Saying he felt a "mix of satisfaction and pride" at having reached the presidency, Viola urged Argentina's youth to "seek participation in the country's political life, so that in the immediate future they can really be architects of a real democracy in our country."

Precisely when and how the democratic restoration will come, though, remains one of the imposing questions Viola inherits. Although the violence of the mid-1970s has been stopped, and improving relations with the United States recently warmed even more with Viola's visit to Washington and New York, the new president is taking power in a country beset by uncertainty.

The economy is in what critics say is one of the worst crises in its history. Negotiators at the Vatican are trying to stave off war between Argentina and Chile. Recent efforts at rapproachement with the political parties have met with decidedly cool receptions, and officials are still being pressured for some accounting of the thousands of people who apparently were kidnaped or killed during the first years of the military government.

"The important thing is for the process to continue," said Videla in his televised farewell address to the nation last week, using the government's code word for its effort to reshape Argentina into an economically successful, subversion-free nation of "Western, Christian values." Declaring that his government had been forced to "confront institutionally the gravest crisis the nation has ever endured," as a result of the previous Peronist rule. Videla spoke with pride of his five years in the presidency.

"Upon assuming control of the government in March 1976, we received a prostrate country, one which with the daily effort of all Argentines, we succeeded in standing on its feet," Videla said. "And today, with pride, we hand over a country that is marching forward. We took a country that was in chaos, and today, we deliver it in order; a country that was near anarchy, which today we deliver with authority; a country that was stagnating, which today we deliver with growing progress; a country that was violent, which today we deliver in peace."

It is known, for example, that although Viola agrees in principle with the previous economic team's goal of "opening" Argentina to increased foreign trade and competition from imports, he thinks too much power was concentrated in the hands of former economics minister Jose Alfredo Martinez de Hoz. When Viola named his Cabinet, he broke up the economics ministry into five separate ministries, some of them headed by ministers who can be expected to take issue with parts of Martinez de Hoz' controversial program.

Worry over the new economic team's direction appears to have fueled the crisis of recent months. Foreign reserved are still dropping as Argentines convert their pesos into dollars to protect against possible devaluation, and the interest on loans has now reached an annual rate in some cases of 365 percent.

Viola is also known as a politically cagey man, who had helped maneuver some fierce hard-liners into the background during the government's late 1970s campaign to wipe out the guerrilla and clandestine left. He reportedly has tried to open conversations with the two major political parties, the Peronists and the Radicals, but party leaders -- in what one newspaper headlined The Boycott of Viola -- so far have responded by asking their members to reject participation in "the process."

Viola's recent five-day visit to Washington and New York, in which he met with President Reagan, Secretary of State Alexander M. Haig Jr., U.N. Secretary General Kurt Waldheim, banker David Rockefeller and Henry Kissinger, was widely viewed here as a triumph.

The Reagan administration, as part of its effort to shift concern over human-rights violations to what it describes as "private diplomacy," has asked Congress to lift the arms embargo that was imposed against Argentina in 1978 when the State Department described it as a "gross and consistent violator of human rights."