The International Automobile Federation (FIA) did not say why Budkowski, a Polish-born aerodynamicist who has previously worked for Ferrari and McLaren, had resigned or where he was going next.
A spokesman said at the Malaysian Grand Prix on Thursday that his duties would be assumed by race director Charlie Whiting.
'Gardening leave' is a term used to describe the time that someone privy to sensitive information must serve before working for anyone else.
The FIA and Formula One's new owners Liberty Media are looking at ways of making the sport more competitive, with a decision also to be made on what sort of engine should be used after 2020.
(Reporting by Alan Baldwin in London, editing by Toby Davis)
