The Dutch driver Jos Verstappen had asked the F1 team to decide on his future by a Friday deadline.
Verstappen had started the race level on points with Hamilton but 9-8 ahead on wins, and there were fears a collision could settle the outcome.
Red Bull's triple World champion Max Verstappen claimed pole position on Saturday for the Miami Grand Prix, to start from the front of the grid for the sixth straight race this season.
Verstappen suffered two retirements from the first three races but has otherwise finished all but two rounds in the top three.
Max Verstappen leads Red Bull front row sweep in Bahrain
Factbox on Dutch driver Max Verstappen, who became Formula One's youngest race winner, aged 18, at the Spanish Grand Prix on Sunday.
Max Verstappen was compared to Formula One greats Ayrton Senna and Michael Schumacher on Sunday after the Dutch teenager's sensational drive to third place in a wet and chaotic Brazilian Grand Prix. Even father and former racer Jos, the 19-year-old Red Bull driver's biggest fan and fiercest critic, was astonished. "What Max showed here today was really special," he said of a race likely to go down in the sport's annals as one for the ages. In the last 15 laps, after a change to extreme wet tyres, Verstappen passed more than half the starting grid as he went from 16th place to the podium.
Former Ferrari technical director Ross Brawn says 70th Anniversary race winner Max Verstappen reminds him of Formula One great Michael Schumacher. Brawn masterminded Schumacher's seven world championships with Benetton (1994 and 1995) and Ferrari (2000-04) and was team boss at Mercedes when the German made his comeback from 2010-2012.
Lewis Hamilton has hailed Max Verstappen's stunning success in Formula One and called on those criticising the Dutch teenager's driving to keep a sense of perspective.
Max Verstappen's sensational first Formula One victory, at just 18 years of age, had former champions and grandees of the sport casting around for superlatives at the Spanish Grand Prix on Sunday.
A weekend that began at the top of the timesheets for Max Verstappen ended with the Dutchman out of the points and in a war of words with Ferrari rivals after a nightmare Belgian Formula One Grand Prix.
Red Bull driver Max Verstappen has received a flurry of criticism from the Formula One world, including legendary driver Niki Lauda who asked him to consult a psychiatrist after the teenage driver was involved in a three-car collision in last week's Belgian Grand Prix.
Max Verstappen gave himself a perfect 20th birthday present by winning the last Malaysian Grand Prix on Sunday, while Lewis Hamilton stretched his Formula One world championship lead to 34 points.
Dutch rookie Max Verstappen, signed controversially as a 16-year-old by Toro Rosso, has swept a hat-trick of awards after confounding his critics in an impressive debut Formula One season.
Dutch teenager Max Verstappen became Formula One's youngest race winner on Sunday in a sensational Spanish Grand Prix that saw dominant Mercedes drivers Lewis Hamilton and Nico Rosberg collide on the opening lap.
Formula One world champion Lewis Hamilton was fastest but teenage rookie Max Verstappen stood out with an eye-catching practice performance on a rainy Thursday at the Monaco Grand Prix.
Dutch teenager Max Verstappen is a rare talent comparable to the late Ayrton Senna, according to Red Bull's Formula One consultant Helmut Marko.
Luca di Montezemolo's resignation as Ferrari chairman on Wednesday severs an historic link with the company's late founder Enzo and marks the end of an era for Formula One's most successful and glamorous team.
The first 17-year-old to race in Formula One became the sport's first 18-year-old driver on Wednesday and Toro Rosso's Max Verstappen is happy to reach the milestone.
New boy Vettel plays down Ferrari hopes in Melbourne.
Michael Schumacher's son Mick is '100 percent' certain to make it to Formula One and when he does it will be great for the sport, world champion Lewis Hamilton said.
2016 is at the halfway stage and the year has already seen some stunning sporting wins, underdogs emerging triumphant on the biggest stages of them all.
