WASHINGTON (AP) _ Thousands of jubilant Republicans celebrated George Bush's brand-new presidency at 11 inaugural balls Friday night, and during a fast-paced tour of the merrymaking Bush took a few reluctant whirls around the dance floor.

He called himself ''a lousy dancer'' and told one crowd rocking to the music of Lionel Hampton: ''I hope you don't judge my being a president by the way I dance.''

A chill wind whipped evening gowns and fur stoles as 60,000 party faithful thronged from Union Station to the National Air and Space Museum to the Kennedy Center on the banks of the moonlit Potomac for a night of dancing and carousing.

Inside the Kennedy Center, the air was hot and thick as 5,000 guests were jammed shoulder-to-shoulder in the Grand Foyer to awaiting the president's arrival. Several people fainted, and for one woman, the crowd was so dense that she couldn't fall down after she passed out.

Hours after he was sworn in as President Reagan's successor on the steps of the Capitol, Bush began his tour of the invitation-only balls with a visit to an American Legion affair for 149 Medal of Honor winners - including freshman Sen. Bob Kerrey, D-Neb. - and other military heroes.

''I'll do my level best to be a president who respects the military ... and keeps the United States of America strong,'' said Bush, who was accompanied by former Rep. Edward Derwinski, R-Ill., his choice to be secretary of the new Department of Veterans Affairs.

Bush also let slip that he's not particularly keen on dancing. ''This is the first formal event ... and perhaps the most humane,'' he said, ''a ball at which you do not have to dance.''

But by the time he reached Union Station, the president finally took his wife, Barbara, onto the dance floor as the band struck up ''I Could Have Danced All Night.''

''You can say that you saw it first here,'' Bush said, ''a lousy dancer trying to dance his first dance with the first lady of the United States of America.''

A while later, at the cavernous Pension Building, the president quipped, ''the Bush family is going to give dancing a bad name.'' He took a few turns with Mrs. Bush and stopped abruptly. ''That's it,'' he said. ''That's the only step I know.''

Vice President Dan Quayle apparently felt the same way. He told revelers at another hotel ballroom that ''I guess protocol allows you to witness my inability to dance.'' Quayle then took a few stiff turns on the dance floor with his wife, Marilyn, as the band played a tune from ''My Fair Lady.''

Tickets to the nine crowded black-tie balls sold for $175 each, assuring guests the privilege of jostling for drinks at cash bars and catching brief glimpses of Bush and Quayle. Tickets were easier t obtain for two casual-dress balls for the 30-and-under crowd, where the drinks had more fizz than kick.

''To be included in this is incredible,'' said Kerry Trost, 16, of Fairfax, Va., who got her $30 ticket from a friend and joined several thousand other youngsters in dancing to loud recorded rock music and videos.

When Bush showed up at one of the youth balls, he recalled the inaugural parade and said, ''It was the young people that brought that thing alive.''

Outside one hotel, a group protesting the killing of animals for their fur hides picketed arriving guests, many of whom wore fur coats. The demonstrators carried signs reading, ''Fur Is Murder'' and ''I'm Warm. I'm Not Wearing Fur.''

A star of one hotel ball was 7-year-old Jessica Catonia, the current Little Miss Buffalo. The girl's mother, Sharon, said her daughter was conceived at the same hotel during Reagan's first inauguration in 1981.

Before the hoopla began, inaugural officials had said a $175 ticket to the black-tie events, reserved for invited guests, would get them little more than ''the honor of being there.'' Bars at each site were charging for drinks, there was no food and the dance floors promised to be gridlocked.

The partying continues Saturday with scores of social events around the capital, including a ''Black Tie and Boots'' ball for 5,500 Texans that sold out within 24 hours.

The four days of inaugural festivities costing a record $22 million to $25 million will end with a Sunday morning worship service at Washington Cathedral which was requested by Bush.

One highlight of the week was a two-hour variety show Thursday night at the Washington Convention Center, where Bush and Quayle joined 12,000 black-tie guests in applauding big-name performers from Frank Sinatra to country singer Loretta Lynn.

The gala was broadcast live on television and grossed about $9 million from ticket and TV commercial sales to help defray costs of the privately financed inauguration week.

Meanwhile, those protesting the policies of the Reagan administration and what they fear will be those of the Bush administration held their own Counter-Inaugural Ball at the Crispus Attucks Museum on Friday night, with a $6 to $10 sliding scale admission charge.

The dance party was arranged by the Counter-Inaugural Coalition for a People's Agenda, a coalition of 80 groups ranging from the Gray Panthers to the Student Coalition Against Apartheid and Racism.

Those attending were asked to wear ''their finest trash,'' said Michael Hasty of the Washington Peace Center, a member of the coalition. Live music was planned for the first hour, followed by a disk jockey playing tapes ''so people can dance the night away.''