Geographically:
The Red Sea lies to the west of both Saudi Arabia and Yemen, separating them from Africa (Sudan and Eritrea).
The Gulf of Aden lies to the south of Yemen, connecting the Arabian Sea with the Red Sea through the Bab el-Mandeb Strait.
The Gulf of Oman lies to the east of Oman, connecting to the Arabian Sea and Persian Gulf — not to the west of Saudi Arabia or Yemen.
The Black Sea is far away, in Eastern Europe, and irrelevant here.
The Persian Gulf lies to the east of Saudi Arabia, not the west.
Therefore, the body of water directly to the west of Yemen and Saudi Arabia is the Red Sea.
\[
\boxed{\text{Red Sea}}
\]