White has been part of the coaching set-up at Hampshire since 2012 and took on head-coaching responsibilities in July when Dale Benkenstein left the club.
White, who made 30 Test and 51 one-day international appearances for England, captained Yorkshire to promotion from Division Two of the County Championship in 2005 and has had his position confirmed at the Ageas Bowl.
He said: “I’m proud and honoured to be offered the role. I want to create a relaxed and focused environment where the players are in the best frame of mind possible to apply themselves on the field and perform moving forward.
“I’ll be doing my very best to ensure that the club, its players and supporters enjoy success in the coming years.”
Kagiso Rabada claimed five wickets to lead South Africa to a 177-run victory over Australia in the first Test at Perth.
Australia put up a brave fight chasing a notional 539 for victory with half-centuries from Usman Khawaja and Peter Nevill but ultimately fell short, bowled out just before tea on the final day for 361 when Keshav Maharaj trapped Nathan Lyon lbw.
Rabada shouldered the load after Dale Steyn was sidelined by a shoulder injury on the second day and returned 5-92 for the fourth five-wicket Test haul of his career after Australia began the final day on 169 -4.
England paceman Harry Gurney has signed a two-year contract extension with Nottinghamshire.
The 30-year-old, who joined his home county from Leicestershire in 2012, took 81 wickets in all competitions in 2016 and his contract was due to expire at the end of next season. He has now committed his future to the club until the end of 2019.
Gurney, who has played 10 one-day internationals and two Twenty20 internationals for England – all in 2014 – has claimed 306 wickets in his five years at Trent Bridge.