Can a pulmonary embolism be detected on a chest X-ray? If not, how are they diagnosed?
In short, no.
A chest x-ray simply doesn't provide the level of detail to look at the vasculature of the lungs.
Instead, the golden standard for diagnosis of a PE would be a CT angiogram of the chest; in which a radioisotopic contrast is injected into the patient. The radiologist would then examine the vasculature carefully to look for any areas of occlusion - made evident by the contrast which glows white when bombarded by x-rays. So where there is bright white, there is blood flow. Where there isn't, but should be, it will appear dark, demonstrating the presence of an embolism.
Furthermore, there is another type of imaging study called a V/Q (ventilation-perfusion) scan, which again, uses a radioisotope to assess airflow and circulation in the lungs.