2023 Dubai champion Medvedev was beaten 2-6, 7-6(7), 7-5 by Griekspoor, with the world No. 6 now the highest-ranked opponent the Dutchman has beaten in his career.
Victory came for Griekspoor after former world No 1 Medvedev was unable to convert four match points late in the second set, and this loss continues a difficult run of form for the Russian.
The 29-year-old has not won an ATP title since the Italian Open in May 2023 and has not reached a final since Indian Wells in March 2024.
In recent months, Medvedev has been the leading critic of ball speed and quality on the ATP, with the former world No 1 repeatedly expressing his ire.
That proved to be the case once again following his defeat to Griekspoor in Dubai.
"It was ridiculous how slow the balls were," said Medvedev. "I do not understand how it's not 5-7 ball change [after games] in the ATP Tour. I don't get it.
When further asked about how conditions may have aided the serve-and-volley tactic employed by Griekspoor throughout parts of the contest, Medvedev expressed similar frustrations.
He added: "The balls, when they fly through the air slower, get an advantage to guys who volleys because they have more time to react, to adapt to a shot that is going to them when they had more time, and that's a big disadvantage but the ATP doesn't seem to care.
"I think he [Griekspoor] started playing better at some moments. I do think in a way, he got lucky with some shots but that's tennis. When I say lucky, he also went for them so he deserved it.
"I didn't really get tired or anything. I didn't really get tight or anything. I played good shots at the good moment. He made it happen for him today."
The top seed in Dubai, Medvedev's quarter-final defeat means that only one seeded player - fourth seed Stefanos Tsitsipas - is through to the semi-finals of the ATP 500 event.
Two-time tournament runner-up Tsitsipas faces Griekspoor for a place in the semi-final, while in-form Felix Auger-Aliassime faces qualifier Quentin Halys on the bottom half of the draw.
Medvedev will next be in action in Indian Wells, which starts next week.
The former US Open has been beaten in the final the past two years - losing to Carlos Alcaraz in both 2023 and 2024 - and is currently set to be the fifth seed at the prestigious Masters 1000 event.