In this study, the normal stiffness of elastic contact between rough surfaces with asperities following Gaussian distribution is investigated using ubiquitiform theory, developed from fractal theory. In the generalized ubiquitiformal Sierpinski carpet model, the rough surface including contact asperity is controlled for, given the lower bound to scale invariance of rough surfaces. Considering the stiffness of a single asperity deduced from the Hertz contact model, we deduce the theoretical relation between the normal stiffness and the elastic contact of rough surfaces based on ubiquitiform theory. The results show that the normal contact stiffness of a rough surface increases as the normal load rises. If the ubiquitiformal complexity of a rough surface increases or the lower bound to scale invariance of a rough surface decreases, the normal contact stiffness of the rough surface should increase. The larger the ubiquitiformal complexity of a rough surface is, the more obvious the impact of the lower bound to scale invariance on the normal contact stiffness of the rough surface becomes. The results based on the ubiquitiformal model and the experimental results are in closer agreement. Therefore, the introduction of scale invariance is crucial to the surface contact problem.