The ex-Northampton boss helped Graham Henry and Steve Hansen guide his countrymen to World Cup glory late last year and has since taken up a role with Super 15 side the Waikato Chiefs.
And Smith insists he wants to honour his commitment to his new club, although he admits that coaching England would interest him further down the line.
"I have a moral obligation to fulfill my time here with the Chiefs," Smith, who also questioned what the role of England head coach would actually involve, told the Dominion Post newspaper.
"The job description was waffly and looked like it was for the head of a PR company rather than a rugby coach.
"I'm interested down the track if it fits in and they want me to have an involvement in talking to them.
"I want to coach at that level again and that's a team that has got a fair bit of potential, but this opportunity just didn't fit in with me."
With Smith now no longer an option for the Rugby Football Union, Nick Mallett could become an even more attractive alternative.
Mallett has just finished coaching Italy and has held the reigns with South Africa and Stade Francais.
Smith's lack of interest may also help interim coach Stuart Lancaster, who was placed in charge for the current 6 Nations campaign.
Lancaster, who previously coached the England Saxons and has also worked at Leeds, was appointed following Johnson's decision to not to apply for a contract extension at the end of a disappointing World Cup campaign.
The 42-year-old has led his country to two wins from two games so far in the 6 Nations, with a 13-6 victory over Scotland followed by a 19-15 success against Italy.