Hearts owner Ann Budge has been left "disillusioned and bitterly disappointed" by the vote to end the Scottish season below the Premiership due to the coronavirus pandemic.
Dundee reversed an earlier intention to reject the Scottish Professional Football League's contentious proposals on Wednesday, meaning the resolution achieved the numbers required for the motion to pass.
The vote means the Championship, League One and League Two campaigns will be curtailed early and decided on a points-per-game basis, with Dundee United, Raith Rovers and Cove Rangers declared champions.
Partick Thistle and Stranraer will be relegated under the proposals, which could see bottom-placed Premiership club Hearts also drop, should the SPFL board decide the top-flight season cannot resume.
In a lengthy statement on the club website on Thursday, Budge said: "Yesterday afternoon's announcement that Dundee FC has re-cast its vote, in support of the SPFL's written resolution of 8th April, leaves me both disillusioned and bitterly disappointed."
She added: "It is now of paramount importance that we do not give up in our efforts to right this wrong."
Partick Thistle, facing relegation from the Championship, have already sought legal action while Stranraer, who will go down from League One, described the decision as "grossly unfair".
Budge, who will lead a task force looking at a potential restructuring of the top-flight with Hamilton chairman Les Gray, said the process had been badly handled.
"I have said from the outset that it was imperative that we find a solution that ensures no club is penalised by relegation in a situation where the leagues cannot be finished," she said. "The financial consequences are enormous."
She added: "This is what must now be addressed through a temporary change to the league structure to cope with what is likely to be two very 'abnormal' seasons."
Clubs facing relegation could still be saved by further talks over league reconstruction.
A popular proposal among clubs and fans would see two teams promoted to the top four leagues and no side relegated.