The Real Madrid midfielder, who suffered a hairline fracture of a rib against Wales, would have missed Wednesday's game through suspension anyway after picking up a yellow card on Saturday.
He was forced to withdraw from the coxless four when he sustained a stress fracture of the rib, leaving him with a deflated lung, and could only watch on as the crew claimed gold without him.
An official at the hospital said the man was in critical condition with multiple rib fractures, and that his female companion was also being treated there for a pelvis fracture.