Can I connect a combination heater/fan unit to the existing bathroom wiring?
In this case, it looks like you'll have to run a new 20A circuit.  As per Article 110.3(B) of the National Electrical Code (NEC), and the manufacturers instructions.

NEC 2008
110.3 Examination, Identification, Installation, and Use of Equipment.
(B) Installation and Use. Listed or labeled equipment shall be installed and used in accordance with any instructions included in the
listing or labeling.

On page 5 of the manufacturers installation manual (PDF), it clearly states that you must use a 20A circuit.