Mr Stilwell unfortunately labored under the misconception that rockets have to "push against" something to fly. They do not. Rockets operate by the equal but opposite law in that they move forward by "throwing" huge amounts of mass backwards. That mass is composed of gasses, but those gasses are moving at several times the speed of sound and gather much momentum thereby.
The earth, not being "pushed against" does not change its spin or orientation due to a rocket launch. (If it did, it would be infinitesimally unmeasurable due to the difference in mass between the planet and rocket.)
