Van den Boog, a director since January 1 2009, had a public falling out with Cruyff last week when he refused to execute a technical plan from the European Cup winner's advisory board.
In Cruyff's plan manager Frank de Boer, Dennis Bergkamp and Wim Jonk should be in charge of all technical decisions involving the first team, youth department and scouting.
"All developments around the report of the technical advisory board had an enormous impact on everyone in our organisation, but also for me personally," Van den Boog said.
"It made it impossible for me as accountable director to take decisions involving the near future and therefore I can't fulfil my duty like I should."
Earlier chairman Uri Coronel gave up his seat on the supervisory board in support of Van den Boog after the clash.
"It is a very sad decision but we respect it. Van den Boog did a great job for Ajax and therefore it is painful that all the matters had such a big impact also on him personally," said Coronel.
Ajax won their last championship in 2004, but after Sunday's 2-1 win at NEC Nijmegen moved them one point behind leaders PSV Eindhoven and Twente Enschede with three matches remaining they are still chasing their 30th title.
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