Mr McMillan is to step down at the end of the year, although the CBI insists his departure was planned before the embarrassing fiasco that has gripped the group for more than a week.
However, Yes campaigners have suggested Mr McMillan had been blamed for the debacle that saw 18 organisations, including Scottish Enterprise, universities and the BBC either quit the body or suspend their membership.
A Nationalist MSP last night added his name to those calling on Mr Cridland to step down as UK leader of the CBI over his role in the organisation’s public opposition to a Yes vote.
SNP MSP Chic Brodie criticised Mr Cridland’s handling of the controversy, after the CBI boss suggested a junior employee had filed forms with the anti-independence Better Together campaign without permission from senior officials.
Mr Brodie said: “John Cridland really must consider his position in the CBI. Blaming a junior member of staff for the foolish decision to register with the Electoral Commission as an official backer of the anti-independence No campaign is cowardly and just won’t wash.”
Mr Cridland previously stated that the bulk of the CBI membership do not think the case for independence has been made and that Scotland and the “economy of the United Kingdom is stronger together”.
However, Tony Banks, leader of the pro-independence Business for Scotland campaign, claimed Mr Cridland had been the “main driver” in the CBI’s initial decision to register with the Electoral Commission to formally back a no vote in the referendum.
Mr Banks accused Mr Cridland of running a “highly politicised” No campaign, before the CBI reversed its position.
He said: “Mr McMillan’s retirement as director in Scotland may suggest to some people an element of scapegoating to distract from the central CBI figure, director-general John Cridland. His position now looks untenable.”
Mr Banks also suggested that the CBI’s whole input in the referendum debate was now compromised because of Mr Cridland’s initial backing for a No vote.
He said: “The news that Mr McMillan will soon be departing from CBI Scotland comes as no surprise. His silence during the last week speaks volumes.
“However, it does beg some questions about the main driver behind the CBI’s response to the Scottish Government’s white paper.
That was headed up by the director-general, John Cridland, who was responsible for the composition and media promotion of what was clearly a highly politicised act of No campaigning.”
The CBI initially registered with the Electoral Commission, allowing it to spend up to £150,000 in campaigning for a No vote ahead of the referendum on 18 September.
Mr Cridland has insisted that the CBI was “politically independent and impartial” when the body faced criticism over its referendum intervention, with groups such as the Law Society quitting the body.
The CBI has said that Mr McMillan, 63, who was appointed in 1995, agreed a departure timetable for the end of 2014 at the beginning of the year. The organisation last night restated its claims that it was a non-political body as it defended the activities of its employees.
A CBI spokesman said: “The CBI has made its position clear. As businesses work hard to secure the economic recovery, the CBI and all its staff have a job to do on behalf of CBI members and their employees to help create the right conditions for UK companies to grow and prosper, wherever they operate around the world. We will continue to do that without fear or favour.”