Mr Staechelin said that he made the sale to diversify his family trust's investments, which are largely tied up in a collection of 26 valuable impressionist and post-impressionist works amassed of by his grandfather.
Experts said that the sale could be the start of a new trend of trophy sales that bypass auction houses in favour of lucrative direct deals.
Nafea Faa Ipoipo depicts two young women in bright colours before a landscape - one in native dress, the other in a colonial, missionary style dress - symbolising European convention and Polynesian custom.
Gauguin painted the work during his first visit to Tahiti, aged 43-44, where he travelled to escape "everything that is artificial and conventional" in Europe.
He returned to Polynesia in 1895, where he remained until his death from an overdose of morphine at the age of 54 in 1903.
Auction sales made a total of $16.1 billion in 2014, according to New York-based researcher Artnet.
Mr Staechlin has confessed to never having hung the paintings from his grandfather's collection in his own home because they were too valuable.
"In a way it is sad," Mr Staechelin told the New York Times. "But on the other hand, it is a fact of life. Private collections are like private persons. They don't live forever.