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.
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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.