What was the reason for his (peace be upon him) looking from top to bottom, then lowering his head, and why did he not marry her after a woman offered herself to him?

The Prophet, peace and blessings be upon him, looked at the woman who offered herself to him, raising and lowering his gaze. This indicates the permissibility of looking at and scrutinizing a prospective fiancée. If she pleases him, he may marry her; otherwise, he may decline her. This is what the Prophet, peace and blessings be upon him, did; he looked at her but did not desire to marry her.