Sir David currently works three days a week at HS2 and will remain in his role until a replacement is found.
HS2 is also currently looking for a new chief executive after Simon Kirby left in September to join Rolls Royce.
Sir David said: “Whilst the case for Gatwick expansion in the future remains very strong, the challenge for Gatwick now is to continue to invest so it can maximise the use of its existing facilities and so can do even more for Britain in the coming years.
“I look forward to working closely with Stewart Wingate and the Senior Executive Team. Their impressive track record to date provides a strong foundation for the future.”