Who else watches the captain's welfare?

Is the executive officer the third person responsible for looking after the captain?

The additional person identified is the ship's senior petty officer. The executive officer's main responsibility is to look after the crew below.

The captain checks on the executive officer, and the executive officer looks after the people below. Responsibility for the captain's own welfare therefore falls to a few people whose role allows them to look upward in the hierarchy. The ship's senior petty officer is added to the steward and medical officer in this explanation.