@Background Pony #58A7  
Uh.. no? I’ve done it to many images and it’s always worked perfectly, no sharp edges.  
Did you even try it? It takes like 30 seconds and works on any image with black outlines and white background.  
If anything, your method of “setting white pixels to transparent” would not include any pixels that are not completely white, which there are a LOT of around anti-aliased lines.  
If it were pixel art with binary edges, sure, that’d be a solution, but not for stuff like this.
 
My method is to ensure that the gradient of pixels from white to black on smooth edges are set to appropriate opacity, it would convert pixels from this:  
opaque -> opaque -> opaque  
black -> grey -> white  
to this:  
opaque -> semitransparent -> completely transparent  
Black -> black -> black(but invisible so colour is irrelevant)

@Background Pony #3884  
Just set all white pixels to 100% transparency and fill the eyes back in. Your method is messy and would leave sharp edges.

>select section  
>expand selection by 2 or 3 pixels  
>click “set brightness to opacity”  
>???

@Background Pony #3884  
Magic.

How did you manage to make the image smaller in the process?