At the beginning of Chapter 11, you will come to a point where you see Riggs trapped under a burning vehicle. You have a gun with one bullet. You can either choose to shoot him and put him out of his misery, or you can walk away and let him burn.
For this achievement, just walk away and leave him burning under the car. You can reload checkpoint and get the other decision or wait for your 2nd playthrough of campaign.
NOTE: If you have not yet gotten the Deer Hunter achievement, now is an almost unmissable time to get it. Since you chose not to shoot him, your hand gun still has one bullet in it. As your walking away, you will see an oryx right in your path out in the open. Shoot him and the Deer hunter achievement is yours.