Can an F-16 perform Pugachev's Cobra?
No.

This would require a docile pitch behavior of the airframe up to approx. 110° angle of attack. The design of both the Su-27 and the MiG-29 is based on a geometry that has been carefully optimized by TsAGI to behave nicely over its full AoA range, a care that has not been spent by General Dynamics while designing the F-16. The specifications did not call for it, because the idea was to create a low-cost lightweight fighter to complement the heavy F-15.

However, even the F-15 is AoA limited because less care was spent on its aerodynamic refinement. At higher angles of attack, both the F-16 and the F-15 show a nasty pitch-up which cannot be controlled with the aerodynamic control surfaces - only thrust vectoring can manage this. Contrast this which the Su-27, which shows balanced and slightly negative (nose down) pitch moments over the full AoA range, so the limited control authority coupled with the aircraft's dynamics can make the Cobra maneuver possible.