Some agencies' (a few that I know of in my state) policies do require the officer(s) to provide medical assistance to a subject they have shot if EMS hasn't arrived, and, IF in the officer's judgement, it is safe to do so.
As for a private citizen, one suspects that is is up to the individual -- personally, if I have shot someone inside my home, verified there are no additional suspects inside (have good security camera coverage) and it is safe to approach the suspect (assuming another family member is home and could provide cover), I would at least apply a dressing to the wound(s) and attempt to stop further blood loss ... after 911 has been notified and EMS is on the way (the deputies and the fire medics are typically at least 15 - 20 minutes away from my semi-rural dwelling).
It is my moral obligation to provide at least basic first aid (I do have combat medical training) and prevent loss of life if possible, to anyone, no matter the circumstances. As far as a legal obligation, if I'm off-duty, I can't speak to that ...
Our state does have a "Good Samaritan" law to protect those who do attempt live-saving measures. And certainly, it could speak to the righteousness of a shooting and one's good intentions if one performed medical aid subsequent to having to use lethal force.
But eventually, after every shooting, one is judged in the court of public opinion. See the Tulsa PD case (Crutcher shooting) after the officers stood around while the subject bled out.