After you escape with Grant, Vaas will find you and shoot Grant.
Now you will have to complete a Quick-Time Event in order to heal him.
Spoilers
When Vaas shoots Grant, there is no way to help him.
Grant's wound is in his neck, and even if you complete the QTE successfully (which should be no problem - the player simply has to hold down one button for a pre-set duration) he will bleed out and you will be unsuccessful in your objective.
The only thing left to do now is run far away from Vaas' camp.
