Coulter Osborne had been appointed by Labour Minister Lisa Raitt.
CUPW said the resignation creates an opportunity for both sides to return to the bargaining table and negotiate a deal.
Union president Denis Lemelin said the union has asked for a meeting with the minister.
"We will encourage Ms. Raitt to appoint an experienced and bilingual mediator to work with the parties to negotiate a collective agreement instead of replacing the arbitrator," Lemelin said in a statement.
Canada Post locked out some 50,000 of its employees earlier this year after a series of rotating strikes by the union.
The dispute was ended after federal back-to-work legislation passed in June forced workers to accept wages that amounted to less than Canada Post's last offer.
On other issues, the law imposed a form of winner-take-all arbitration in which both the union and the post office make a final offer, one of which would be accepted.
Canada Post said it would look for direction from the minister and remained committed to the process.
"We are disappointed by the delay, but remain hopeful for a resolution that will address Canada Post's structural cost issues and improve the corporation’s long-term sustainability," Canada Post said in a statement.
Raitt encouraged the two sides to reach a negotiated settlement on Thursday, but noted the legislation requires that an arbitrator be appointed.
"Canadians gave our government a strong mandate and for the good of the economy and all Canadians this matter needs to be settled," she said.
The union had launched a court challenge to Osborne being chosen as the arbitrator because they believed he lacked the necessary experience in labour relations and, as a unilingual anglophone, would not understand francophone witnesses.
A federal court judge had agreed to put the arbitration proceedings on hold ahead of full arguments on the case in January.
The union is also mounting a separate legal challenge of the constitutionality of the legislation that forced its members back to work.
The government has insisted that the intervention was required to protect a shaky economy.