A GPS cannot do this.
It only reports how fast the plane is actually moving, which says nothing directly about how much airflow the pilot has to work with. In addition, it figures in wind effects, which also have no direct usefulness in controlling the plane.
Pop quiz:
You are at 35,000 feet. The GPS says you are traveling at 400 kts due north over the ground, decreasing at 1 kt per second. There is a 60-kt jet stream blowing from the southwest (217°). Your stall speed (no flaps) would be 167 kts INDICATED, but your airspeed indicator isn't working. Your heading is 356°. Barometric pressure is 29.75 (but your altimeter is set to 29.92 as in all flights above 18,000 feet.) The air outside is -42° C.
Not absolutely true.
for a given configuration, GPS(s) systems can be used as a sole source to give attitude and performance data, including AoA. (20 years ago... in experimental testing, nowdays my cat has a GPS...) In fact they can even be used to track wing bending/fuselage torsion etc... if you use the carrier wave rather than only the signal directly.
At least on the B777 TBC came out with a simple switch, the FPV which just gives the derived FP, and is a ready indicator of AoA for a... er..."crosscheck". (S&L it is a great analogue of AoA, but in high bank angles it takes a little more thought to determine the AoA from the display). The underlying source is hybrid AoA/ADC through the ADIRU, which of course removes the opportunity for redundancy.... maybe next time.
The question is how conditions that were encountered by a first world operator, operating state of the art technology, all generally conforming with the structured, bloated,bureaucratic, costly guidelines of EASA/JAA defeated all the safety protocols so spectacularly.
AF447 crew response is not isolated, and that should cause concern for those passengers that consider dummying down aerospace to a commodity product as packaged and sold by M.O.L., Safety is expensive, but the public pocket has driven the industry to the point where it is argued that a MPPL is a good thing, that pilots are operators not pilots.
Next time you fly, look out the window, that thing out there is a wing. XBox, MS FS etc don't usually employ such things. Using a computer simulation, normally you don't end up dead, as the m.v^2's are generally of a much low order of magnitude. There is nothing trivial about the kinetics of an aluminium tube stuffed with assorted people moving in 3d space (4d... avoids MAC's) at high speeds, with lots of hydrocarbons, yet, as indicated elsewhere we are happy to pay the taxi driver more for 15 minutes, or the terminal company more for 1:00 hr of parking than we pay for the opportunity to dance around the sky.
As Feynman said in '86,
"For a successful technology, reality must take precedence over public relations, for nature cannot be fooled."
PS:
the impact of auto trim from erroneous speed information, and the control law reversion needing the cognitive engagement of the crew to the dynamics of the situation in a highly stressful sudden onset event, that is additionally time critical is much easier to quarterback later, with the benefit/certainty of hindsight. (AI products have had numerous events and disasters where the trim system has resulted in an out of trim condition). I do not criticise the crew, I criticise the bureaucratic system and public apathy that results in conditions precedent where lives are needlessly lost in the pursuit of cheap travel. Pay in cash or blood, simple choices.