Harry arrived in a Mercedes EQC electric car driven by six-time world champion Hamilton and was greeted by dignitaries including former world champion driver Damon Hill.
The pair also met F1 drivers George Russell and Alex Albon, along with local schoolchildren.
Silverstone was an airbase during World War II, and the first British Grand Prix of the modern era was held there in 1948.
On Thursday, Harry and his wife, Meghan, the duke and duchess of Sussex, attended an awards ceremony for wounded service personnel, appearing together for the first time since January's announcement that they would stop their royal duties.
Next week, Harry, 35 and Meghan, 38, will carry out what is expected to be their last official appearance alongside other senior royals including the 93-year-old queen, Harry's father and heir-to-the-throne Prince Charles, and his brother Prince William at the annual Commonwealth Service at London's Westminster Abbey.