I fell on the right side of my right wrist, pain with movement, almost no swelling, extremely low range of motion.

Need x-rays to : Rule out fracture, which clinically is associated with more persistant pain/swelling/lrom.

Fracture?: Common injuries as a result of such a fall with limited swelling are a scaphoid fracture causing pain on the radial [thumb] side of the wrist, or a tfcc tear that would cause pain on the ulnar side of the wrist. You need to see a hand surgeon and obtain further studies ie. X-ray.