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 £35million 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 football 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.
"He has been very open and honest with me from minute one. He has inherited a very difficult situation and it is just how he sees the best way out of it. The discussion about my future will take care of itself."
He added: "I like the group of lads. I've got on with them very good and when George Boateng, Kevin Kilbane, players of that ilk speak well of you, there is a good buzz.
"I like the area, it sits well with me, but the decision is not about me, it's about the club being solvent and us going forward."
Hull fell behind after half an hour at the DW Stadium when Victor Moses curled in a fine effort off the post, his first goal 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 Premier League accounts, but Gohouri - another Latics first-time scorer - dealt Hull a cruel final blow with an overhead kick at the death.
It denied the Tigers what would have been a morale-boosting first away victory of the campaign, after Dowie's decision to field a youthful side - including Atkinson, Cullen and Tom Cairney from the start - almost paid off.
"That is the nature of this harsh league," Dowie said.
"To put it into perspective, we've put three lads in there and I felt they gave us an injection of enthusiasm and energy, and a bit of a cutting edge.
"I'm very proud of the display, I thought it was absolutely outstanding and that we deserved more than we got.
"We haven't won today but we might well have done. We have scored two goals away from home for the second time since I have been here and unfortunately we have only got one point from it, which is the galling side."
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.
"It is one of those performances which leaves you with mixed feelings because we knew that we had to take control of the game and break Hull down.
"They were well-organised and working hard and I thought for spells, we showed very good moments.
"You score the first goal and you feel you are going to kick on, but then we conceded two goals that were very soft and that is a clear area that we have to improve for next season.
"But then what we don't lack is that character and desire to carry on doing the right things until the end. I thought the second goal was the reward the players deserved."