Lewchew - Japan relations remain poor and tense due to the long history of prejudice, discrimination, and genocide being committed by Japan against Lewchewans into the present day. 
Lewchew maintained friendly relations with both China and Korea, and certainly had no desire or incentive to do anything that might jeopardize those relations. The request by Japan for Lewchew to send troops must have seemed laughable to Lewchew. It was ignored.
Lewchew continued to be recognized as a prosperous, peaceful, sovereign country by the other countries of the world, including Japan.
During the Ryukyu Disposition of the 1870s Japan confiscated the treaties Lewchew had signed with foreign nations, such as the U.S., France, and the Netherlands. The Government of Japan holds these documents to this day, and refuses to return them to Lewchew, despite repeated requests. 
The government of Japan refuses to recognize Loochooans as indigenous because it would essentially mean that Loochoo would regain its independence, and Japan would no longer be able to oppress Loochooans and control military bases in Loochoo.
In 2008 the United Nations Human Rights Committee stated that Japan should recognize Ryukyuans as indigenous.
In 2009 UNESCO recognized a number of Ryukyuan languages, as well as Ryukyu's unique ethnicity, history, language, and traditions, thought Japan still refuses to recognize these things. 
In 2010 the United Nations Committee on the Elimination of Racial Discrimination (CERD) stated that they regret "the approach of the state party [Japan] to accord due recognition to Okinawa's distinctness and expresses its concern about the persistent discrimination suffered by the people of Okinawa.  It further reiterates the analysis of the special rapporteur on contemporary forms of racism that the disproportionate concentration of military bases on Okinawa has a negative impact on residents' enjoyment of economic, social and cultural rights."

Prehistory 

The Japanese archipelago was originally populated by the Ainu, dating back to around 30,000 years ago. This is known today as the Jomon period. 
DNA evidence suggests a close genetic link between Loochooans and Ainu.  
Between the years 1000 B.C. through 300 C.E., waves of migrants from other parts of Asia (most likely Korea and China) came to Japan. They intermixed with or killed off much of the Ainu, and the Ainu were gradually pushed further and further northward. This group is known today as the Yayoi people, or modern Japanese. Modern Japanese average around 97% Yayoi ancestry

Bibliography 


  1. Gakuhari, Takashi, et. al. "Jomon genome sheds light on East Asian population history." https://www.biorxiv.org/content/biorxiv/early/2019/03/15/579177.full.pdf
  2. Kanzawa-Kiriyama, Hideaki; Kryukov, Kirill; Jinam, Timothy A; Hosomichi, Kazuyoshi; Saso, Aiko; Suwa, Gen; Ueda, Shintaroh; Yoneda, Minoru; Tajima, Atsushi. "A partial nuclear genome of the Jomons who lived 3000 years ago in Fukushima, Japan". Journal of Human Genetics. February 2017. 62 (2): 213–221.