By the very definition offered in the article:
Geo means Earth and Patho means suffering. It means suffering from the earth which causes stress in the form of illness.
The Earth has no nervous system and is not scientifically known to suffer, although we can certainly imagine it doing so. That's a feature of human psychology, which is what this boils down to: psychosomatic illness.
The locus classicus for "geopathic stress" is a 1993 article by an alternative medicine practitioner named Jane Thurnell-Read. She defines it as "the general term for energies emanating from the earth that cause discomfort and ill health in human beings," and compares it to concepts such as ley lines, dowsing, and feng shui.
To back up her original terminology, Thurnell-Read cites a Weimar period dowser named Gustav Freiherr von Pohl as well as an article called "Cancer in the light of geophysical radiation" in a 1927 issue of an American journal called Cancer. Neither of these people used the term "geopathic stress". Furthermore, common sense would tell us that the earth is not a dangerous substance that emanates radiation so much as a large, solid thing that intercepts radiation, which is why atomic bomb shelters and nuclear waste storage facilities are located underground. [edit: As pointed out in comments above, the earth does emit heat and very low levels of gamma radiation. Heat is more concentrated in some parts of the earth than others but generally it is described as heat and not as "geopathic stress".]
There could possibly be "energies" emanating from the earth unknown to science, that somehow remain reliably in place for years despite plate tectonics and other underground shifts, and by other unknown factors cause discomfort and stress in human beings, but neither Thurnell-Read nor anyone else has explained the mechanism by which this might work, which prevents anyone from falsifying her claim. She admits, "I have used case studies for illustration throughout this book. Unfortunately these do not convince the scientific mind. When I talk to my scientific friends, they tell me I am citing anecdotal evidence." (Geopathic Stress & Subtle Energy, 2006)
By the admission of the person who invented the term, this is not a falsifiable claim, and is therefore pseudoscience. People who claim that certain arrangements of objects or types of electromagnetic waves make them sick are far more likely to have psychological stresses, for which dowsing, feng shui or other practices may serve as unscientific therapy.