The Fairfax County Board of Supervisors gave final approval yesterday to the county's first $1 billion budget, which will maintain current tax rates and freeze most county programs at current levels.

The board also formally abandoned its controversial plan to withdraw 6,880 county employes from the Social Security system, an attempt initiated more than a year ago in an effort to combat the costs of rising payroll taxes and a recession-related financial squeeze.

In its budget for the year beginning July 1, Fairfax was able to avoid any rise in property taxes -- set at $1.47 per $100 valuation -- by forgoing major program expansions, except for day-care centers, new libraries and a new police and rescue computer system.

One controversial cost-saving move by the county was its decision to limit county employes to 3 percent pay raises. Fairfax teachers angrily protested the cap and have vowed to replace some of the current supervisors in this fall's elections with candidates who will support higher pay for teachers.

The tax rate and appropriation figures were approved unanimously by the board yesterday without any controversy.

The move to retain Social Security coverage for county workers, however, was largely symbolic since legislation signed last week by President Reagan effectively eliminated the county's right to pull out of the Social Security system. The board unanimously approved a motion by Supervisor Audrey Moore, an Annandale Democrat, to cancel its notice to the Social Security Administration sent on April 30, 1982, that it intended to withdraw from the system at the end of 1984.

Fairfax, like numerous other financially pressed governments and private organizations, believed it might be able to run a cheaper and more efficient retirement system than the federal government.

In November 1981 the board set up a study group to examine the issue. Five months later, when the board learned that federal legislation might remove that option, they submitted the formal notice of withdrawal to the Social Security Administration. Last December the subcommittee recommended that the county stay in the system, but that the withdrawal notice not be canceled immediately.

The new federal legislation forbids any additional county or state governments from withdrawing from the system, according to an agency spokesman. Because Fairfax was not scheduled to withdraw until next year, it would not qualify, he said. Before the deadline, 981 state and local governments withdrew, according to the agency.