The two British soldiers were dragged from their vehicle after driving into the path of an IRA funeral in west Belfast.
They were brutally beaten before being taken to waste land, stripped and shot.
At the latest monthly council meeting, the DUP proposed remembering the two men.
But the Alliance Party tabled an amendment, proposing that all victims of the Troubles be remembered instead of singling out particular cases.
The Alliance Party proposed a form of words that placed innocent soldiers on the same level as terrorists.
Christopher Stalford, DUP
"It is clear that the Alliance Party has become so consumed by hatred towards unionists that they will oppose almost anything that unionists propose," DUP alderman Christopher Stalford said.
"The Alliance Party needs to take a long, hard look at themselves following this meeting and decide just what has happened to their moral compass."
The DUP accused the Alliance Party of "riding to the rescue of Sinn Féin" by tabling the amendment and said they had "lost their way" by failing to condemn the corporals' murders.
The party insists that its original motion was not controversial or designed to offend anyone.
"It was designed to show respect for Corporals Wood and Howes, to praise their bravery and restraint in the face of a violent mob," Mr Stalford added.
"And to assure their families that their sacrifice would never be forgotten."
Alliance Councillor Mervyn Jones insisted that he had condemned the killings in his speech to the council, but felt that "by having a motion condemning all murders, it would help us move forward".
With many anniversaries coming up in the next couple of years, I believe it is important for the Council to have direction in how we deal with the past by condemning violence from all quarters.
Mervyn Jones, Alliance
He explained: "Alliance proposed this amendment so that Belfast Council would show its opposition to all forms of violence and condemn all the deaths that occurred during the Troubles.
"If we are to work towards a shared society, then we must have an agreed comprehensive method of dealing with the past - instead of concentrating on specific cases.
"We could have motions each month that brought up every incident which would not lead to progress in how we deal with the past."
The mob attack on the two corporals was captured by television cameras and is widely regarded as one of the most chilling moments of the Troubles.
Catholic priest Father Alec Reid witnessed the scenes and tried to intervene to save the men, despite being told he would be shot for his efforts.
He was dragged away, but managed to follow the black taxi which took the soldiers away to be killed. While unable to prevent their murders, Father Reid administered the last rites.