"The End of Summer", a novelette by Algis Budrys which was also the answer to the old question "Memory suitcase" story; first published in Astounding Science Fiction, November 1954, available at the Internet Archive. You may have read it in one of these compilations.
ISFDB synopsis:
A man develops immortality for humans and other animals at the cost of anterograde amnesia, which is overcome by people recording their memories every day and replaying them again the next day.
He accidentally runs over a dog, but it's not his:
He tried to snap the car out of the way, his face suddenly white, but the dog moved unpredictably, its abrupt yell of pain louder than the scream of Fay's brakes. He felt the soft bump, and then his foot jerked away from the clutch and the car stalled convulsively. Even with his engine dead and the car still, he heard no further sound from the dog.
Then he saw the Homebody boy running toward him up the road, and the expression of his face changed from shocked unpleasantness to remorseful regret. He sighed and climbed out of the car clumsily, trying to think of something to say.
The guy who ran over the dog is responsible for creating the network of satellites that blanket the Earth in some kind of radiation that makes everybody immortal but has bad side effects such as amnesia.
