I thought it was by Heinlein or Brown, but if it was I can't locate it online.
You couldn't locate it because it's not Heinlein or Brown, it's Ray Bradbury's "Kaleidoscope", also the answer to this question. A radio adaptation was episode 48 (Sept. 15, 1951) of Dimension X which is available at the Internet Archive.
The plot was aboard a spaceship, or maybe a space station, when an accident sends the astronauts spinning out of control towards certain doom.
The first concussion cut the rocket up the side with a giant can opener. The men were thrown into space like a dozen wriggling silverfish. They were scattered into a dark sea; and the ship, in a million pieces, went on, a meteor swarm seeking a lost sun.
One part of the story I remember distinctly was the one of the guys started screaming inconsolably and the noise was driving everyone crazy.
Now, as if they had discovered the horror, two of the men began to scream. In a nightmare, Hollis saw one of them float by, very near, screaming and screaming.
"Stop it!" The man was almost at his fingertips, screaming insanely. He would never stop. He would go on screaming for a million miles, as long as he was in radio range, disturbing all of them, making it impossible for them to talk to one another.
One of the astronauts made his way over to the guy and smashed his helmet so the screaming stopped.
Hollis reached out. It was best this way. He made the extra effort and touched the man. He grasped the man's ankle and pulled himself up along the body until he reached the head. The man screamed and clawed frantically, like a drowning swimmer. The screaming filled the universe.
One way or the other, thought Hollis. The moon or Earth or meteors will kill him, so why not now?
He smashed the man's glass mask with his iron fist. The screaming stopped. He pushed off from the body and let it spin away on its own course, falling.
As I recall, the story ends with the narrator turning off his comm link so he can ponder the beauty of the universe in silence.
No, they just lose contact as they fly off in different directions:
Now all the voices were fading, each on his own trajectory, some to Mars, others into farthest space. And Hollis himself . . . He looked down. He, of all the others, was going back to Earth alone. [. . .] The voices faded and now all of space was silent. Hollis was alone, falling.
They were all alone. Their voices had died like echoes of the words of God spoken and vibrating in the starred deep. There went the captain to the Moon; there Stone with the meteor swarm; there Stimson; there Applegate toward Pluto; there Smith and Turner and Underwood and all the rest, the shards of the kaleidoscope that had formed a thinking pattern for so long, hurled apart.
The ending:
He fell swiftly, like a bullet, like a pebble, like an iron weight, objective, objective all of the time now, not sad or happy or anything, but only wishing he could do a good thing now that everyone was gone, a good thing for just himself to know about.
When I hit the atmosphere, I'll burn like a meteor.
"I wonder," he said, "if anyone'll see me?"
The small boy on the country road looked up and screamed. "Look, Mom, look! A falling star!"
"Make a wish," said his mother. "Make a wish."
