This statement goes at the heart of Newton's third law of physics, for every action there is an equal and opposite reaction.  When a normal person does a push up, they exert a force on the Earth and the Earth exerts an equal force on them which pushes that person up.  For the forces to be equal and opposite, they must hold to Newton's second law F=ma.  If the average person were to do a push up, they accelerate the Earth down, but the earth is so massive that it hardly moves.  It is its massive size that keeps the Earth from moving when people move about on the blue planet.

While the original statement is true for everyone, as everyone moves the earth by nearly immeasurably small amounts, for the claim to hold any significant water it must be assumed that Norris causes the Earth to move in a noticable fashion.  or Chuck Norris, he legitimately does push the earth down.  That is to say that the earth accelerates very quickly away from him.  In order for this to make sense, his mass must be drastically larger than that of the Earth's.  That is, he is much larger than 5.97 x 1024 kg.