Pressure groups are calling on Viacom and its subsidiary BET (Black Entertainment Television) to pull the video after it premiered on the TV network on Tuesday.
The song is sung from the perspective of a girl who has shot down a man who raped her.
In the first few seconds of the video, Rihanna takes aim at her attacker and shoots him dead in the middle of a crowded street.
Lyrics to the song include: "Momma, I just shot a man dead … I never been so proud."
Paul Porter, co-founder of US think tank Industry Ears, said: "'Man Down' is an inexcusable, shock-only, shoot-and-kill theme song.
"In my 30 years of viewing BET, I have never witnessed such a cold, calculated execution of murder in primetime.
"If Chris Brown shot a woman in his new video and BET premiered it, the world would stop. Rihanna should not get a pass."
Melissa Henson, of the Parents Television Council, said: "Instead of telling victims they should seek help, Rihanna released a music video that gives retaliation in the form of premeditated murder the imprimatur of acceptability.
"The message of the video could not be more off base."
A spokeswoman for Mothers Against Violence said: "Murder happens all around us. We need solutions, not songs influencing vulnerable individuals. Sadly, these kind of scenes are a reality for some."