If we are talking about a single past action, I would say: "He did what he should have done."
I would say that if it was a story about one particular time in which the man stopped to help someone in need.
If we are talking about something he did in the past and we expect him to again do in the future, I would say: "He did what he should do."
For instance, I would say this if his job is to help people, or if I am talking about my son, and I am pleased that he knows what is the right thing to do now and in the future.
I am not certain that other people will agree with me.
A side note: We would say He saw an injured man.Injured is the adjective form; injury is a noun.
It's an interesting question. I'm sure that we've discovered differences between AmE and BrE on similar questions in the past, though I can't put my finger on them at present. From memory, BrE was more inclined to use the whole set of auxiliary verbs than AmE was.
Your current example seems to me to raise a slightly different issue.
Personally, I feel rather uncomfortable with both He did what he should do/He did what he should. However, I wouldn't feel uncomfortable with any of: He did what he felt he should do;
He did what he felt he should;
He did what he had to;
He did what he had to do.
That said, I'm happy with both He did what he should have and He did what he should have done - though I'm more likely to say the second.