The Tigers, who were already as good as down due to their vastly inferior goal difference, had their fate sealed at Wigan when Steven Gohouri fired home an acrobatic effort in stoppage time to level the match at 2-2.
Hull are also saddled with debts of around £35m (€40.4m) and face a summer of upheaval as they attempt to keep the club solvent.
Dowie, who replaced Phil Brown at the helm in March on a short-term deal, is yet to discuss his future with club chairman Adam Pearson, but insists he is happy to wait.
Asked if he would be with Hull in the Championship, Dowie said: “That’s for people much more erudite than me to talk about. I don’t know.
“It doesn’t need to be discussed (yet). Adam knows how to run a club.
“He has been involved for a long time and he is dealing with what are the pressing (issues), which are the financial difficulties.
“It’s important for going forward that the club’s solvency is dealt with first. Mine or any player’s future will become secondary to that.’’
Hull fell behind after half an hour when Victor Moses curled in a fine effort off the post, his first for Wigan since joining from Crystal Palace.
The visitors looked to have turned the match around as youngsters Will Atkinson and Mark Cullen also opened their league accounts, but Gohouri – another Latics first-time scorer – dealt Hull a cruel blow with an overhead kick at the death.
Wigan boss Roberto Martinez, whose side’s top flight status was preserved last weekend, was pleased to see his players salvage a draw in their final home game of the season.
“I felt we did (deserve it), but I think we were to blame for their two goals,” Martinez said. And with Wigan playing Chelsea next weekend, Martinez said: “What we don’t lack is character and desire to carry on doing the right things until the end.”
WIGAN: Stojkovic, Melchiot, Caldwell, Gohouri, Figueroa, Diame (Scharner 82), McCarthy (Sinclair 65), Gomez (Scotland 79), Watson, Moses, Rodallega.
HULL: Duke, Mendy, Mouyokolo, Gardner, Dawson, Atkinson, Cairney, Boateng, Kilbane, Vennegoor of Hesselink (Folan 70), Cullen.
Referee: Phil Dowd (Staffordshire).