The 47th-ranked Dutchman came through 2-6, 7-6 (9/7), 7-5 as the world number six imploded.
Medvedev, the 2023 champion in Dubai, was handed a code violation for unsportsmanlike behaviour after squandering four match points in the second set.
As his frustration boiled over, he accused umpire Adel Nour of applying "double standards" with Russian players.
Nour hit back, telling Medvedev that he treats all players in the same way, regardless of nationality before brushing away the top seed's complaints.
Griekspoor kept his cool in the deciding set, taking victory on a fourth match point of his own.
Medvedev refused to shake hands with Nour at the end of the tie.
The 28-year-old Griekspoor will face either Stefanos Tsitsipas or Matteo Berrettini for a place in the final.