"We have a council to run, and a city to run."
Councillor Rob Pascoe asked if criticism of an elected member, was the same as an attack on an elected member.
Mr King would not answer, saying he was not going to get into the debate publicly.
Councillor Angela O'Leary said she had a problem with the "sweepingness" of the report and asked if the councillors in question had been spoken to.
Mr King again said he was not going to debate it publicly.
"I don't want to get diverted by people trying to get media attention."
"I am not going to talk about things that have happened individually in this chamber.
Later in debate Mr King said that he was referring to more than one incident.
"This is far more than one complaint that has been laid against one member to another."
"This is about your behaviour on social media, and in your commentary."
After the meeting, some councillors apologised on social media for their recent behaviour.
Councillor Geoff Taylor, who had a code of conduct complaint laid against him by Mr Pascoe said he was not going to get drawn into petty debates again.
"The mayor is absolutely right with this stuff. I'm proud of some of our achievements and this stuff doesn't do us any good. Lesson learnt," Mr Taylor said on Facebook.
"Comments I made in a recent media article about Rob Pascoe were incorrect. It's not my style and it's not the way I want to work at the council. I look forward to getting on and working alongside Rob who I rate highly as a colleague and getting good things done for the city."
Mr Pascoe followed suit, saying he wishes for the same and to focus over the next six months in concluding some of the good work from this council term.
"One thing we can agree upon though is that it's the issues we debate, not the individual," Mr Pascoe said on his Facebook.
Hamilton News has asked Mr Pascoe if he will now be withdrawing his complaint against Mr Taylor.
Councillor James Casson, who in January, targeted the mayor and a group of councillors over alleged bloc voting tactics said he agreed with the message, but not the process of how it was done.
"Fair enough and I accept this. I did not vote to receive the report as I personally think we should have been named to remove the implication everyone was involved. I'm moving on and can work with anyone including Geoff. We don't see eye to eye on a few things but have agreed on other items. I would like to think we have had our vent and can now work together for the betterment of our city," Mr Casson said.
Ms O'Leary said on Facebook she did not support the Mr King's report, saying council has a code of conduct process that should had been used to deal with complaints.
"An unfortunate start to the final 9 months of this term. I didn't support King's report and don't accept being admonished by him for a public spat that had nothing to do with me. We have a Code of Conduct and that is the process that should be used to deal with complaints. Let's get on with the business at hand," Ms O'Leary said.