Inada was born in Nagoya and graduated from Tokyo Imperial University in medicine before travelling abroad for medical studies in Germany.

Returning to Japan from Europe, Inada became the first professor of medicine at Fukuoka Medical College (福岡医科大学, Fukuoka Ika Daigaku) of the Kyoto Imperial University (京都帝国大学, Kyōto Teikoku Daigaku), now Kyushu University, School of Medicine.

In 1914–1915, Inada discovered the spirochete bacteria that causes infectious jaundice (Weil's disease) and developed a successful antiserum treatment for the infection.

The initial specimen material (stock of Ictero No.1) which Dr. Inada isolated in 1914 has been preserved as a significant artifact in the history of medicine.

In 1919, Inada and his co-worker Yutaka Ido were nominated by Louis Martin of Pasteur Institute, Paris for the Nobel Prize in Medicine for their 1915 discovery.