Are fully extended TQFTs generalized cohomology theories?

A fully extended tqft is not quite a generalized homology theory... But almost. You can find a preliminary reference here (notes of a talk by Hiro Tannaka at the mit Talbot workshop): http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology(hiro).pdf The precise statements you might be interested in are Theorems 2.16 and 2.20.

(side remark: the notes of the whole workshop are worth reading: http://math.mit.edu/conferences/talbot/2012/notes/talbot_2012_notes(claudia).pdf).

EDIT : the work announced in the above talk is now partly available on John Francis' webpage: http://www.math.northwestern.edu/~jnkf/writ/ (see "Factorization homology of topological manifolds" and "Structured singular manifolds and factorization homology").