SAN FRANCISCO—Attorneys for Uber Technologies Inc. have successfully persuaded a private arbitrator that a California driver for the transportation company is an independent contractor, not an employee, in the first arbitration in the United States to test that issue.

The decision has limited ramifications for the company and the many legal battles it is facing in courts across the nation over the employment status of its drivers. But it provides the first glimpse of how private jurists may approach the issue in the wake of an appeals court ruling that will, if it stands, ensure that most driver claims are arbitrated.