Your question indeed very clear,a good example, and cuts to the heart of physics - being able to define, predict the world using mathematical models. But you have to make sure your models are correct - and complete.
Fabrice and Robert are right - velocity will go to infinity without constraints. And there is the clue as to what's going on as a physics interpretation - constraints. The equation you cite is the steady state solution - velocity of a small mass accelerating towards a larger body with gravitational acceleration g, with a projected area, $A$, and in an atmosphere with density $\rho$.
Remove the atmosphere, and as you say there is no longer a drag force, and so the initial dynamic model from which you derived this steady state formula is different. But when you let $\rho$ go to zero it's (almost) giving you the right answer. I say almost because the equation you cite is not complete without an additional equation: one that constrains the path of the attracting bodies to the point at which they collide - a hard limit. Things can't fall forever - they eventually hit the ground. That's the real and complete physics of the matter.
A caveat: "Things can't fall forever" this might not be true for a black hole. Time and space are so warped it may take 'forever' for the object to reach the singularity according to clocks outside the black hole. But these physics require additional mathematics to model the behavior.