COVID-19 is a pandemic disease, which massively affected human lives in more than 200 countries. Caused by the coronavirus SARS-CoV-2, this acute respiratory illness affects the human lungs and can easily spread from person to person. Since the disease heavily affects human lungs, analyzing the X-ray images of the lungs may prove to be a powerful tool for disease investigation. In this research, we use the information contained within the complex structures of X-ray images between the cases of COVID-19 and other respiratory diseases, whereas the case of healthy lungs is taken as the reference point. To analyze X-ray images, we benefit from the concept of Shannon’s entropy and fractal theory. Shannon’s entropy is directly related to the amount of information contained within the X-ray images in question, whereas fractal theory is used to analyze the complexity of these images. The results, obtained in this study, show that the method of fractal analysis can detect the level of infection among different respiratory diseases and that COVID-19 has the worst effect on the human lungs. In other words, the complexity of X-ray images is proportional to the severity of the respiratory disease. The method of analysis, employed in this study, can be used even further to analyze how COVID-19 progresses in affected patients.