This is probably "Thunder and Roses" by Theodore Sturgeon. From this Reader's Guide to the story:
Pete Mawser and Sonny are survivors of one-half of a nuclear war, stuck on a military base, both plagued by thoughts of suicide. They discover a formerly hidden door that has unlocked due to the high-level of radiation. They get spooked and leave. Starr Anthim is a popular singer but gives a performance without being flashy. She sings the "thunder and roses" of the title.
The story opens with the sense of defeat:
Everyone used to say, Wednesday morning, "How about the way Starr sang The Breeze and I last night?"
That was a while ago, before the attack, before all those people were dead, before the country was dead. Starr Anthim—an institution, like Crosby, like Duse, like Jenny Lind, like the Statue of Liberty. (Liberty had been one of the first to get it, her bronze beauty volatilized, radio-activated, and even now being carried about in vagrant winds, spreading over the earth . . . )
Pete Mawser grunted and forced his thoughts away from the drifting, poisonous fragments of a blasted liberty. Hate was first. Hate was ubiquitous, like the increasing blue glow in the air at night, like the tension that hung over the base.
Their side is doomed:
"You—well, it's not easy to forget. We got hit. We got hit everywhere at once. All the big cities are gone. We got it from both sides. We got too much. The air is becoming radioactive. We'll all—" He checked himself.
The song Starr sings, to try to convince people not to retaliate is:
With thunder I smote the earth
With roses I won the right
With the sea I washed, and with clay I built
And the world was a place of light
She then makes her pitch:
"We have more bombs than both of them put together. We have them. We are not going to use them. Wait!" She raised her hands suddenly, as if she could see into each man's face. They sank back, tense.
"So saturated is the atmosphere with Carbon Fourteen that all of us in this hemisphere are going to die. Don't be afraid to say it. Don't be afraid to think it. It is a truth, and it must be faced. As the transmutation effect spreads from the ruins of our cities, the air will become increasingly radioactive, and then we must die. In months, in a year or so, the effect will be strong overseas. Most of the people there will die too. None will escape completely. A worse thing will come to them than anything they have given us, because there will be a wave of horror and madness which is impossible to us. We are merely going to die. They will live and burn and sicken, and the children that will be born to them—" She shook her head, and her lower lip grew full. She visibly pulled herself together.
She gazed briefly at each man in turn, from the screen. "We must not strike back. Mankind is about to go through a hell of his own making. We can be vengeful—or merciful, if you like—and let go with the hundreds of bombs we have. That would sterilize the planet so that not a microbe, not a blade of grass could escape, and nothing new could grow. We would reduce the earth to a bald thing, dead and deadly.
"No—it just won't do. We can't do it.
Pete Mawser hunts down Starr, where she admits one of the reasons she's spreading her message:
"All right," she said, with a tired acquiescence that frightened something inside him. "You seem to have guessed right, though. It's true. There are master firing keys for the launching sites. We have located and dismantled all but two. It's very likely that one of the two was vaporized. The other one is—lost."
They lost track of where that is, so they're spreading the message to try to convince whoever does find it not to activate it and let it disarm itself. Of course, because it's a story, it happens to be at that very base, Pete and one other man already found it (although both not quite realizing what it was until putting two and two together with the performance)... Pete wants to honor Starr's ideals, while the other man wants to retaliate not caring if that damns the planet.
The complete story can be read here.