Gillinwater, who was appointed to the board just two years ago, will step down from the post at the next annual general meeting in June.
Today, he alluded to the supermarket's recent struggles to compete with discounters.
I have had a thoroughly rewarding few years on the board... notwithstanding the considerable recent upheaval in the supermarket industry. However, my appointment as chairman of SSE, coupled with my other responsibilities, will require a significant time commitment.
The supermarket's management has undergone a radical upheaval over the past couple of months, as it sought to reposition itself.
In January new chairman Andrew Higginson took over from Sir Ian Gibson.