Long-term resident status, granted in consideration of special circumstances, has a maximum designated term of five years.
According to the suit, the couple have been together for around 15 years and married in 2015 following the legalization of same-sex marriage by the US Supreme Court across all 50 states.
In Japan, such unions are not legally recognized.
High has applied unsuccessfully for a Japanese long-term resident visa five times since 2018.
He currently resides in Japan on a temporary visa and argues he could face deportation if the Japanese government refuses to renew it.
The Immigration Services Agency of Japan said it will respond appropriately after examining the details of the suit.
Earlier this year, a Taiwanese man became the first foreign gay partner of a Japanese citizen to be granted special resident status by the Justice Ministry, following the revocation of a deportation order for overstaying his visa.