Nautilus is written in C. I have not found any reference to stat syscall in the source code, so cannot say with 100% certainty that it does or does not use it.
Mime type of files is obtained via nautilus_file_get_mime_type function, which is referenced a lot in multiple files such as nautilus-mime-actions.c , nautilus-properties-window.c , nautilus-window-slots.c and others.
However, i could not find the definition of that function, only assignments:
$ grep -iR 'file_get_mime_type'
nautilus-canvas-view-container.c: mime_type = nautilus_file_get_mime_type (file);
nautilus-mime-actions.c: mime_type = nautilus_file_get_mime_type (file);
nautilus-mime-actions.c: mime_type_a = nautilus_file_get_mime_type (file_a);
nautilus-mime-actions.c: mime_type_b = nautilus_file_get_mime_type (file_b);
nautilus-mime-actions.c: mime_type = nautilus_file_get_mime_type (file);
nautilus-mime-actions.c: char *mime_type = nautilus_file_get_mime_type (parameters->file);
nautilus-mime-actions.c: mime_type = nautilus_file_get_mime_type (parameters_install->file);
nautilus-mime-actions.c: mime_type = nautilus_file_get_mime_type (parameters_install->file);
nautilus-mime-actions.c: mime_type = nautilus_file_get_mime_type (file);
nautilus-properties-window.c: ret = g_list_append (ret, nautilus_file_get_mime_type (NAUTILUS_FILE (l->data)));
nautilus-properties-window.c: mime_type = nautilus_file_get_mime_type (file);
nautilus-properties-window.c: mime_type = nautilus_file_get_mime_type (target_file);
nautilus-window-slot.c: mimetype = nautilus_file_get_mime_type (file);