I can't find any evidence that Sukiyaki was actually in American Gigolo; IMDB doesn't list it.
However, it would almost certainly have been A Taste of Honey, who had a hit with their cover of Ue o Muite Areko at about the same time (1980). But a much superior version was recorded by Big Daddy for their The Best of album.
Sam Cooke Sang This Version
The song Believe It or Not [Theme from The Greatest American Hero] was sung by Joey Scarbury.
Sam butera
Though I cannot be certain to which song you are referring, it is likely you are thinking of the song 'Sukiyaki' (Known in Japanese as 'ue o muite arukou') by Kyu Sakamoto, which reached number one of the Billboard Top 100 in 1963.
The song, "ABC" was sung by The Jackson Five, an American musical family group. The Jackson Five made the song in 1969 and released it on February 24, 1970.
Sukiyaki - song - was created on 1989-09-14.
Ozone...teenie marie background band
Japan http://www.answers.com/topic/sukiyaki-song
Just a Gigolo - song - was created in 1928.
Gigolo - Mary Wells song - was created in 1981.
Gigolo - Nick Cannon song - was created on 2003-10-18.
Gigolo - Elena Paparizou song - was created on 2006-11-15.
Sam Cooke Sang This Version
Sukiyaki was a number one hit it 1963 by Kyu Sakamoto.
Kyu Sakamoto-1963
Another word for Gigolo (more modern) is Womanizer ^^
The theme song "Secret" is by the American band The Pierces