What Master Chief posted is all true. On the other hand, since you could hear him coughing, you may also be able to slip and shoot him again. Or he may run a short distance and die. You have to do what you think is right according to where you hunt.
Just because we have always heard to back out and wait does not mean that is the right thing to do. Yes, it may result in a longer tracking job but it also accounts for the recovery of many deer that would not be found if you wait.
The right thing to do is whatever you think you should do. That is what I always do and yes, I have lost a few deer. However, I cannot remember the last one I lost and I have recovered a few that I know I would not, had I let them rest. You can bet your bippie, if I think I have one lung shot single or double, I am going to push that sucker as hard as I can. I want him gasping for breath and sucking in blood if I can push him that hard.
Of course...I'll be gasping just as hard. :)