My roommate left giving only 7 days notice and says he is only required to pay the prorated portion of the month he was still there.Is it true that no notice of leaving is required if there is no written lease?

Your roommate is wrong. If there is no written lease then they are a month-to-month tenant and they must give 30 days written notice.

Yep , he is on a month to month rental, and as such a 30 day notice is required