Why don't TAWS systems provide automatic avoidance?
I would have to disagree with most of what John K has said. I will try to provide my own answer as well.


  Theoretically possible, but it would have to be a very expensive and complex sort of system, because it would have to manage pitch and thrust together, meaning it would have to integrate an autothrottle function in addition to an autopilot function


Current generation fly-by-wire aircraft have massive amounts of computing power and are capable of linking together autothrottle and autopilot actions already. Yes, it's expensive, but modern airliners are.


  the risk level of that sort of incident, which are pretty rare outliers really when you think of millions of hours flown each year, doesn't justify the development costs.


Indeed, thankfully, all accident types are enormously rare in modern aviation. However, that doesn't mean we stop trying to improve and CFIT (controlled flight into terrain) accidents are actually some of the more common and certainly warrant the development of safer systems.

My answer: As ymb1 has shown, this system does exist already. Why is it not on airliners yet? Well, I think it will be soon. But there might be a number of issues. 

Firstly: false positives, where a terrain avoidance command is issued without need. Say, for example, an aircraft is coming into land and passes over a tall tree, this sets off the system needlessly and could very well cause further issues with traffic avoidance and airspace incursions - pilots would be hesitant about allowing such a system.

Secondly, aircraft performance. For the jet fighter in ymb1's video the aircraft performance, it's agility and power, and the pilot protection (e.g., helmet and 5-point harnesses) mean that this type of rapid terrain avoidance command is possible and effective. For any large commercial airliner, those sort of commands are not possible (much lower thrust-to-weight ratios etc.) and so the aircraft would have to respond to a terrain avoidance command a long way out - this would increase the likelihood and issues associated with my first point. Additionally, things are not as secure in an airliner and so there is the very real possibility of hurting people and breaking things when such a command is followed - if this is a do-or-die situation, then who cares! But what if it's a false positive and now you've just injured 50 people for no reason.

I'm sure there are other reasons. But having said all that aircraft designer are looking for solutions for CFIT accidents. Currently, for airliners, the main solution seems to be greater awareness for the pilots. So, using HUD/HGS (Heads up display/ guidance system). Some even now have special cameras at the front so that the surrounding terrain is overlayed on the HUD for use in low-light, low-visibility conditions. I guess that is the current solution for now. So expect to see greater use of HUD systems with terrain overlays in airliner cockpits.