The Eredivisie club had announced in March that the former Manchester United and Netherlands international defender would succeed ex-Ibrox midfielder Giovanni van Bronckhorst for the 2019/20 campaign.
Stam leaves with De Stadionclub 12th in the table having won just three matches so far this term. They were beaten 4-0 by Ajax at the weekend.
The 47-year-old admitted it was "certainly not an easy decision", but felt it was for the good of the club.
In a statement on Feyenoord's website, Stam added: "I've thought about it a lot and for a long time.
"My final conclusion is that it is better for the club, the players and myself if I step aside."
Technical director Sjaak Troost said: "The trainer certainly still had our support and the player group also wanted to continue with him.
"But if someone says 'I no longer believe in it and want no more', we respect that and it will stop. What a shame."