Are there any short stories that aren't Americanized that have the theme about cherishing those you love?
The genre of the short story existed before the United States. In fact, much of American fiction is based on European models. Thus, although the United States is one of many countries that produces literary works, it is far from the oldest country in existence, and is far from being a source of all literary traditions.
Many of the authentic tales of love and loyalty from before the existence of the United States (and therefore completely unamericanized) are written in verse rather than prose, and thus are not "short stories," which by their definition are written in prose. If you are interested in narrative poems, with traditional African, Indian, or Middle Eastern settings you might wish to ask a separate question about narrative poems rather than short stories.
Some works you might find interesting include:
Gustave Flaubert (French, 19th century): "A Simple Heart" from Three Tales has a strong theme of love and loyalty.
Alice Munro (contemporary Canadian): Alice Munro is one of the great artists of the short story, and is often considered the quintessential chronicler of the Canadian prairie. Many of her stories are about loyalty and relationships.
Jorge Luis Borges (Argentinian, 20th century): "Emma Zunz" focuses on a woman's loyalty to her father.
Oscar Wilde (British, 19th century): "The Happy Prince" and several of Wilde's other short stories are about forms of selfless love.
Maxim Gorky (Russian, 19th century): "Her Lover." Several of his other stories have themes about loyalty and love.
Anton Chekhov (Russian, 19th century): Several of his stories have love as a theme but tend to be satiric.
Guy de Maupassant (French, 19th century): "The Necklace"