Mr Solomon has threatened to either resign or appeal to the United Nations if the council does not drop its custom of saying a prayer before each meeting.
During Monday's meeting, which at times became heated, Councillor Philippa Baker-Hogan put a motion that the code of conduct be brought back to the table for review.
However, only three councillors voted for the motion - Ms Baker-Hogan, Mr Vinsen and Councillor Ray Stevens - and it was lost.
Most councillors expressed their views that the code of conduct wasn't perfect but it was adequate.
Councillor Sue Westwood said: "We have to have a code of conduct, and no matter whether we change it or not, there will always be loopholes and anomalies."
Councillor Michael Laws said his preference was to have no code of conduct at all.
"We've just spent all this time on a code of conduct hearing and now you want to discuss changing it? I say we should throw it out, but that's just my opinion."
Mr Solomon said he would not support Ms Baker-Hogan's motion.
"I think it might be within the operational discretion of the chief executive and the mayor, at the receipt of a complaint, to investigate whether there might be some way of clearing the water, for the two parties to talk to each other.
"In this situation, I would have been open to that," Mr Solomon said.
Ms Baker-Hogan said she found Mr Solomon's comments "bizarre".
"What he's suggesting actually could not have happened under this code of conduct."
Ms Baker-Hogan later told the Wanganui Chronicle she was disappointed in the council's decision.
"Council has turned down the opportunity to review a code that will cost it $20,000 in this instance, with no confidence at all that we will not be faced with further complaints under this code."
Ms Baker-Hogan said it was concerning that the council had to deal with a code of conduct complaint after only one year.
"Already a new councillor with a poor attendance record has gone ahead with a complaint, not upheld on any count by an independent lawyer or the vast majority of elected members."