I read this in the 90s, but I remember that, when I mentioned it to my mother, she commented that it was a fairly well-known book which she herself had read when she was younger (she would have been in her late 30s when she made that comment, so she probably would have read it in the 60s). Most or all of the drama happens in a hospital, I want say one outside of a major city and thus somewhat constrained in terms of resources, both medicine and personnel.
One of the main characters is a pathologist who is dating a fairly young woman (maybe younger than him). During a tryst, possibly just post-coitus, he's caressing her knee and realizes that something about it doesn't feel right. He has it checked out (perhaps discreetly without letting her know) and finds that she has cancer in her knee, removal of which will probably require amputation of her knee and everything below it. He has a crisis of faith, wondering whether he should tell her himself, and also whether he would have noticed it sooner if he hadn't been distracted by their romance.
There's another plotline going on with a pregnant mother. The Rh factor test either is not performed, or is performed incorrectly, which leads to complications that threaten her life and that of the baby. I don't think the pathologist was directly involved with this case, at least initially.
Lastly, some part of me wants to say that the plot twist, such as it is, is that the death is an older man who was briefly mentioned in the story.
I was initially thinking that the title was Common People or Ordinary People, but I'm coming up with nothing useful, and I think I'm thinking of the Judith Guest novel with the latter.