Councillor McVey was blasted for "shambolic leadership" and for allowing "uncontrolled and unregulated commercialisation" of the Capital's city centre by petition founder, Brian Smith, who also called for the SNP councillor's resignation.
On Twitter, Cllr McVey responded by stating many of the issues which have led to "legitimate criticism" were "avoidable and in some instances simply down to poor communication".
He added that guidance for residents surrounding Hogmanay, where those with homes within the event area were told to apply for access, was "unacceptable" and created "unnecessary confusion".
The petition follows months of controversy over the Christmas Market and the Hogmanay street party, both events run by Underbelly, with calls for the company to be stripped of the contract.
In November it was revealed the Christmas Market in Princes Street Gardens did not have planning permission for a new scaffold structure.
'Disneyfication' and 'over-exploitation of Edinburgh
Petition founder Mr Smith blasted Cllr McVey in his petition, claiming the council was not focused on basic services such as waste and road repairs.
He said: "The over-exploitation, commercialisation of public spaces, and the turning of Edinburgh into a constant uncontrolled theme park lies fairly and squarely with Edinburgh City Council.
"Adam McVey is the leader of this administration and therefore should resign."
The petition cites as reasons for Cllr McVey's resignation the lack of planning permission for the Christmas Market, controversy around how memorial benches were "discarded" during construction, and contractors "throwing aside" the Christmas nativity scene on the Mound.
It also lists the "immense fury" caused by the requirement for residents to pass a check for access to their homes during Hogmanay.
Mr Smith added: "We need someone that will stand up for and protect our city and serve its residents interests. We need somebody who deals with matters and doesn’t bury their head in the sand.
"We need somebody with leadership capabilities who is able to make tough decisions in the public interest. We need somebody who leads Edinburgh into the future in a sustainable manner."
'A citywide conversation' to shape future events
Cllr McVey responded to several of the criticisms on Twitter and said many of the issues encountered this year were "avoidable" or due to "poor communication.
He said: "Earlier this year I confirmed a city-wide conversation in 2020 to help shape future events and meet our aspirations and concerns. There are understandably strong feelings and we have to listen to all residents and respond to meet the challenge.
"Issues with delivery, process, planning, contract management etc. of this year’s events are being considered by councillors in Spring. Many problems which led to legitimate criticism were avoidable and in some instances simply down to poor communication."
Cllr McVey added that a review of the events over the Christmas period will take place in the Spring, and criticised Underbelly's guidance for residents affected by Hogmanay.
He added that there is no "one quick fix" for the issues raised by residents connected to perceived over-tourism in the Capital.
He said: "Residents rights and access to their homes during any event have to be prioritised and properly planned for. It is unacceptable that the guidance for residents didn’t accurately reflect this, creating unnecessary confusion. Any changes required to avoid a repeat will be adopted.
"More generally, many issues being raised are part of wider debate about how we manage size and scale of festivals and tourism. This is why we pushed hard to secure the powers we need to control short term lets and for a tourist tax to provide resources to better manage these impacts
"We shouldn’t fall into thinking that all this can be easily resolved with one quick fix or that everyone shares one view. We’ll publish details shortly of how we’ll engage with city to look afresh and decide collectively what’s best for our city, our residents and our economy."
You can view the petition in full here.