This is a major issue in contemporary ethics. I think that most people make the moral judgment that there is some sort of distinction between doing and allowing although it is difficult to pin down. However, I do not think it has anything to do with greater evil. The right thing to do might be to allow the greater evil to occur rather than doing an evil oneself.
The go-to example would be of a surgeon about to operate on a relatively healthy patient whom the surgeon discovers is a perfect match for four patients in the hospital running out of time for live-saving transplants (of different organs) and finds himself faced with a decision: do I kill the one relatively healthy patient and make it look like a surgical complication and save four lives? Or do I treat and release him and allow the other four patients to (probably) die? Most people think it would be wrong to kill the healthy patient, but not because of any greater goods or evils at issue--obviously four deaths are worse than one.
We disagree. See my essay (a lecture given at Baylor and posted here) "Sin Boldy: Christian Ethics for a Broken World." Some acts are necessary even though wrong (sinful). My argument is that God automatically forgives necessary acts such as taking life to save innocent life.
You are asking me to engage in casuistry. I can only say what I think I would probably do--when talking about "limit cases." And in this case I don't know. But I doubt I would condemn the surgeon in your hypothetical scenario. Have you seen "Sophie's Choice?"