The calculation is entirely correct -- we cannot shoot a projectile into space. There is no sensible way to get something to go fast enough to make it to space from Earth with no further propulsion. A better argument, actually, is about acceleration. In order to not be utterly crushed, a projectile would need a massive vertical track to be shot into space. Earth's escape velocity is around 10km/s. g=10m/s^2. If we assume maximum acceleration of 10g to not crush the pilot, to get out, we'd need to accelerate at 10g for 1,000 seconds. During that time, we'd travel a few thousand kilometers. Accelerating faster -- as with a nuclear explosion -- only exasperates the problem.
We solved the problem by having propulsion the whole way up. We can fire a rocket into space, and it can take it's sweet time to get up there.