I know that now they are appointed by the President and confirmed by the Senate but I can't imagine they allowed George Washington to appoint all of the justices.
Washington nominated all six Justices (the Court originally had only six Justices). John Jay as Chief Justice and John Blair, William Cushing, James Wilson, James Iredell, and John Rutledge as associate Justices.
The Senate quickly confirmed the nominees.
The Judiciary Act of 1789 established the Judiciary branch, and set up a supreme court and circuit courts below it. The Constitution stated that, "judicial power of the United States, shall be vested in one supreme Court, and such inferior Courts." Enabling President Washington to appoint the original 6 justices.