Stokes, 24, reached the milestone from 163 balls on the second day of the second Test in Cape Town on Sunday, obliterating Botham's mark of 220, set against India in 1982.
Analysing the match for Sky Sports, Botham said: "It was magnificent, one of the best examples of Test cricket I've seen in a long time."
Stokes thrashes rapid double century
"Ben Stokes is box office. We knew he'd got it, and this is the tip of the iceberg. He'll get it wrong once in a while, but he is a guy that wins matches, and they don't come along very often.
"You're just king of the castle. No boundary's big enough, no bowler's quick enough. It really is a moment to enjoy."
2nd Test: Late Stokes burst gives England the edge
Stokes's innings was also the second-fastest double ton in Test history, surpassed only by Nathan Astle's record of 153 balls for New Zealand against England in 2002.
Former England opener Geoffrey Boycott said: "All those thousands of England supporters who've come out here for a bit of a holiday, a bit of sunshine, see some cricket -- you've seen something special."
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