Does using a macbook with an external monitor damage the health of the battery?
While the external monitor itself is not the cause of the 'weakened' battery, leaving your MacBook Pro on AC power all of the time will definitely affect the battery's performance.  It is essentially under a constant state of charging, even though this is regulated by the charging circuit, which will eventually cause the battery's chemistry to begin to break down.

Current batteries will typically last the longest when used regularly, but not completely discharged with each use.