Repair fallen stone above fireplace
You will need mortar to resecure the rock.  Here is a good guide as to what product to select.  Check the second item under mortar. 

https://www.lowes.com/projects/build-and-remodel/masonry-and-cement-products-buying-guide/project

As you predicted, you will need to remove most of the d mortar from both the wall and the backside of the stone.  Use a small cold chisel and a hammer.  This is required for a good bond and so that the stone is at the same level as the others. 

Mix a small batch, make sure it's not too wet and smear it on the backside of the stone and on the wall.  Apply the stone and hammer it with a rubber mallet until it is level with the others.  Clean up any mortar that has leaked out. 

Now the hard part, you will need to secure the stone until it dries.  Probably the easiest would be a small piece of wood wedged under the stone to hold it up and use painters tape to hold it against the wall.