Why is Arabian Sea not an ocean?

A sea, as defined by the NOAA, is smaller than an ocean, and partially enclosed by land, not covered on all sides as you say. The Arabian sea fits this description quite well. The boundaries of the Arabian sea, from wikipedia. read more

The Arabian Sea, also known as Sea of Oman, is a region of the northern Indian Ocean bounded on the north by Pakistan and Iran, on the west by the Gulf of Aden, Guardafui Channel and the Arabian Peninsula, and on the east by India. Historically the sea has been known by other names including the Erythraean Sea and the Persian Sea. read more