SELECT COUNT(*) CNT
  FROM (
        SELECT TO_DATE(:END_DT) - LEVEL DT
          FROM DUAL
          CONNECT BY LEVEL <= TO_DATE(:END_DT) - TO_DATE(:ST_DT)
        ) A
 WHERE TO_CHAR(A.DT, 'D') NOT IN ('1', '7')

1. 시작일 기준

SELECT COUNT(*) CNT
  FROM (
        SELECT TO_DATE(:STR_DT) + LEVEL - 1 DT
             , TO_CHAR(TO_DATE(:STR_DT) + LEVEL - 1, 'D') DNM
          FROM DUAL
          CONNECT BY LEVEL <= TO_DATE(:END_DT) - TO_DATE(:STR_DT) + 1
        ) A
 WHERE DNM NOT IN ('1', '7')
 

2. 종료일기준
SELECT COUNT(*) CNT
  FROM (
        SELECT TO_DATE(:END_DT) - LEVEL + 1 DT
             , TO_CHAR(TO_DATE(:END_DT) - LEVEL + 1, 'D') DNM
          FROM DUAL
          CONNECT BY LEVEL <= TO_DATE(:END_DT) - TO_DATE(:STR_DT) + 1
        ) A
 WHERE DNM NOT IN ('1', '7')

구간을 구해야 하기 때문에 반드시 종료일 - 시작일 + 1 의 형태가 되어야 합니다.