Fairfax County Republicans, seeking to buttress their fragile majority on the county Board of Supervisors, scrambled yesterday to capitalize on Democratic Supervisor James M. Scott's decision to resign, setting the stage for a fierce battle with Democrats who said they cannot afford to lose the seat.

"This is a very important race for the Republican Party," said newly installed Fairfax GOP Chairman James D. Swinson. "If we win this seat , then it's going to be some time -- if ever -- before the Democrats regain control of the board."

Harris N. Miller, chairman of the county Democratic committee, said that if the Democrats lose Scott's seat, "it would be that much harder to win a majority" in the 1987 general elections, when all nine county supervisors face reelection. But, he noted, "I don't think we're likely to lose it."

Scott's decision, which was announced formally at yesterday's county board meeting, has surprised Democratic and Republican supervisors alike, setting off a storm of rumors and speculation about who will emerge as candidates for the Providence District seat he has held since 1972.

In a prepared statement that he read to the board, Scott, 47, the most liberal member of the county board, quoted a song by Kenny Rogers: "You've got to know when to hold 'em and know when to fold 'em." He added, "I decided it was time to fold this phase of my life and move on to another."

Board Vice Chairman Martha V. Pennino (D-Centreville), her voice cracking and tears spilling down her cheeks, said she was "dumbfounded" to learn of his resignation.

Scott said he has accepted a newly created job as director of community affairs for the Fairfax Hospital Association, a nonprofit organization that owns Fairfax Hospital and other health-related facilities in Northern Virginia.

The Republicans, who hold a 5-to-4 majority on the board, won control of the Board of Supervisors in November 1984 for the first time in a century.

Many Fairfax Republican officials said yesterday that state Del. Stephen E. Gordy, a 66-year-old former school principal, would be the strongest GOP candidate for Scott's position, which will be filled in a special election this summer.

Gordy, who was elected to the state legislature in 1981, said he is "seriously considering" the candidacy.

In contrast to the usual political progression in many parts of the state and the nation, the Board of Supervisors in Fairfax is considered a step up from the state House of Delegates.

Board members, with an annual salary of $21,589, are paid almost twice the salary of state delegates, represent larger districts with more constituents and generally receive more local media exposure.

Democrats, who generally sounded confident about their chances to hold onto Scott's seat, said Providence School Board member Katherine Hanley now appears to be the party's strongest potential candidate. Hanley, 43, a longtime civic leader, said, "I'm certainly strongly considering" running for the board.

In addition to Gordy, at least two Republicans expressed interest in seeking the job: Robert Beers, an aide to Supervisor Thomas M. Davis III (R-Mason), and Steven A. Armstrong, a lawyer and computer services firm executive who lost to Scott in 1983.

In addition to Hanley, another Democratic possibility is Providence District planning commissioner Rosemarie Annunziata, a lawyer, who said she is considering entering the race.

Scott, who read his three-page resignation statement in an even voice, was showered with praise from his colleagues on the board for his advocacy for human rights, the environment and other issues.