Human doctor cannot travel to help his Martian friend

That is Clifford D. Simak's City story, "Huddling Place". See a good description of the story on Wikipedia. It was first published in Astounding in 1944 -- and it's a Retro Hugo nominee for 2020. It's in pretty much every collection of the City stories.

The story is lightly plotted, and most of the text is devoted to setting the scene. In the distant future, man has colonized Mars and lives an apparently easy life, supported by efficient and intelligent robots. Intelligent Martians co-exist with the humans on that planet. The trend to suburbanization, first manifest in the mid-20th century, has continued, such that many (most?) humans on Earth live in isolated enclaves. Jerome Webster, the main character, is a human with expertise in Martian physiology, especially that of the brain. Like many other human adults, he suffers from progressive agoraphobia, which becomes extreme after his only son departs to spend time on Mars. Jenkins, the most senior family robot, explains to Webster that his father had been similarly afflicted. Webster contemplates writing a monograph on the subject.

Before he can begin this project, he learns that Juwain, a Martian friend and brilliant philosopher, has contracted a terrible disease that only he can cure. This would require traveling to Mars, something his agoraphobia makes nearly impossible. Senior political figures make clear that Juwain's death would be a tragedy for which mankind would suffer for thousands of years, and Webster is pained by the thought of forsaking his friend.

With great effort he packs for the trip, only to discover, in the last lines of the story, that the robot Jenkins, not understanding the stakes, had dismissed the spaceship that had arrived to transport Webster to Mars. The reader is made to understand that Webster probably could not steel himself again for departure, and so Juwain would die.