Why is it appropriate to announce ‘stable’ on final approach?
The FAA offers some advice on this: 


  An unstable approach is simply an approach that does not meet the
  criteria for a stable approach established by the aircraft operator.
  As an illustration, Flight Safety Foundation defines a stable approach
  in the following terms:
  
  On the correct flight path:
  
  
  ILS Approach - ILS within 1 dot of the localizer and glide slope.
  Visual Approach - Wings level at 500 feet AGL.
  Circling Approach - Wings level at 300 feet AGL.
  Only small heading and pitch changes required.
  Speed within +20/-0 kts of reference speed.
  Aircraft must be in proper landing configuration.
  Maximum sink rate of 1,000' per minute.
  Appropriate power settings applied.
  Briefing and checklists complete.
  During IMC - Stable by 1,000 feet AGL.
  During VMC - Stable by 500 feet AGL.
  
  
  If the approach is not stable by 1,000 feet AGL or 500 feet AGL
  (depending on weather conditions), or if the approach becomes unstable
  below these altitudes, the pilot should initiate a missed approach/go
  around. The pilot may initiate a go around at any time above or below
  these altitudes if deemed necessary. It is possible for a pilot to
  initiate a go around even after touchdown on the runway, but not after
  the thrust reversers have been deployed.


They are simply verbally verifying the approach is stable and they don't feel they need to go around. I would think Various OpSpecs dictate how go-arounds are initiated but in reality either pilot can initiate a go-around.  If an approach is unstable a go-around should be initiated. 

IATA advises that callouts are used which may be why carriers are adopting them. 


  Furthermore, the adoption of calls of “STABILIZED”, “UNSTABLE” or
  “GO-AROUND” at a given point on the approach (stabilization
  altitude/height for example) may improve decision making and
  compliance to ensure a timely go-around is carried out. While a
  “STABILIZED” callout might be required at either 1,000 feet or 500
  feet above touchdown, the “GO-AROUND” command can and must be made at
  any time prior to deployment of thrust reversers. Once again, if such
  callouts are adopted it is essential that an acknowledgement is made
  by the other pilot in every case