The story centers on the memories of a writer (Gregory Peck) who is taking a safari in Africa. He develops a dangerous wound from a thorn prick, and lies awaiting his slow death. The loss of physical capability causes him to look inside himself - at his memories of the past years, and how little he has actually accomplished in his writing.
You can stream The Snows of Kilimanjaro (1952) free and legally on OldReel, via the Internet Archive.
Yes — it is in the public domain and can be watched and shared freely.