From the paper:

"The major threat to the internal validity of our findings is that surgeons may selectively operate on sicker and more complex patients on their birthday, perhaps because those patients cannot have their procedures delayed. However, this is unlikely to explain our findings because we found that patients who underwent surgery on the surgeon’s birthday were similar in all observable characteristics to patients who underwent surgery on other days. Furthermore, severity of illness as measured by predicted mortality, and the number of procedures performed per surgeon, also did not differ based on whether a surgery occurred on a surgeon’s birthday compared with other days."

In this case, based on another comment above about "emergency procedure" having multitude of meanings, you're most likely wrong in that the paper has a rebuttal to the top post. The hypothesis then, is that the actual urgency of surgeries is not controlled precisely enough to state that they cannot affect the measurement.