He was raised in a ballet family: his parents were both professional dancers and own their own studio in Tokyo.
In 1980, Horiuchi won the Prix de Lausanne, the international ballet competition in Switzerland.
In 1982, he was invited by George Balanchine to join the New York City Ballet, where he became a principal dancer in 1989.
He played Mr. Mistoffelees in the musical "Cats", both on Broadway and in London's West End.
He choreographed the opening ceremony of the Winter Olympics in Nagano, Japan.