Glasgow Tories believed Cllr Aitken should say sorry after she announced plans for a car-free zone in the city centre before consulting with businesses and residents.
The motion calling for an apology failed to gather enough votes however, with the majority of councillors instead backing an SNP and Green group amendment.
The council leader accused them of personal attacks and said they were showing “total cowardice” by hiding their views behind “fake outrage”.
Cllr Aitken said: “Here we go again. The Tories managed a single full council meeting talking about a better way of doing politics before they are straight back to playing the woman instead of the ball.
“Personally attacking me because I said something in public. They just can’t help themselves.
“The notion that ideas, proposals, intentions, can’t ever be discussed outside a formal consultation or committee process, whether by me or any other elected official of the council is laughable.”
She added it was clear the Tories don’t support the car-free zone plan.
“Instead of having the guts to say so they are hiding behind fake outrage and operational whataboutery. As if solutions for the challenges weren’t found for Buchanan Street many years ago.
“If Paris can turn the Champs-Elysees into a city park, our officers have got the smarts to reprioritise some road space on West Nile Street.”
The Conservative group claimed that Cllr Aitken had been trying to “boost her personal profile” by revealing the policy during COP26.
Bailie Kyle Thornton said no political group on the council has a majority, adding this “isn’t the way to make policy”.
He highlighted comments by Stuart Patrick, chief executive of Glasgow Chamber of Commerce, who expressed his disappointment that there was no prior consultation before the announcement.
Cllr Thornton added: “We have to make sure our policy process is strong. My group believes it is incumbent on the leader of the council to apologise to all those who were caught off guard on this proposal.”
He also said she should apologise for the “sheer disrespect”.
SNP councillor Angus Millar said: “It is of course entirely appropriate for a council leader to state a policy aspiration and to share emerging proposals.”
He added one of his “ key takeaways from COP26 last month is that Glasgow and other cities are acting as a driving force for transformative change for a low carbon future”.
Labour’s Bailie Soryia Siddique said: “Politics has room for disagreement without becoming personal.”
She said her group was “wholeheartedly criticising the administration for its handling” of this situation, but the city has “recognised for a long time that repurposing public space for people rather than cars is a distinct benefit”.
Cllr Siddique added there needs to be consensus and the council must “bring citizens along with us”.
The approved amendment stated “reducing private care use is vital to reduce emissions” and “reallocation of road space from private cars is needed in order to facilitate improved public transport, active travel modes and high quality public realm”.