2 GPa is 2GN/m^2, or 200kN/cm^2, which supports 20Mg. Taking silk density as 1 g/cm^3, that is 20M cm, or 200,000 m, or 20km. Silk density is actually ~1.37 g/cm^3, so derate the above to 14.6 km. IOW, a silk cable can support 14.6 km of rope hanging below it. Not close to strong enough for a Space Elevator.
We will be burning rocket fuel to get to space for the foreseeable future. Better to launch enough to build in-space processing facilities if you're really committed to dual-homing humanity or making space travel more cost effective.