Scotland's ancient universities elect rectors to chair their governing bodies and champion students' interests.
Speaking at his inauguration, TV executive Steve Morrison said the new Higher Education Governance Bill could lead to their removal.
The Scottish government said it had no plans to abolish the role of rector.
The position of rector is an ancient one and in many respects often ceremonial. The rector of Edinburgh University presides over the institution's governing body.
Inevitably, some rectors play a largely symbolic role - in 2014 students at Glasgow University elected whistleblower Edward Snowdon who is currently claiming asylum in Russia - and on occasions, celebrities and public figures have been elected.
Other rectors, however, are in a position to try to be proactive champions of the students' interests.
Rectors are elected at the four ancient universities - Glasgow, Edinburgh, St Andrews and Aberdeen - and also at Dundee. The University of the Highlands and Islands has a rector although the role there is different .
In his inauguration speech, Mr Morrison said he believed other Scottish higher education institutions should follow Edinburgh's example and warned proposed changes to the way universities are governed could end the role of the rector..
One proposal in the bill is to directly elect the chairs of each university's governing body, but some critics claim these elections could involve people other than students and staff.
Mr Morrison said: "Every student and every member of staff, academic and non-academic, has the right to vote for the rector who presides over the governing body of the university - that is our inclusive Edinburgh model.
"I respectfully propose that every Scottish higher education institution should follow this example when electing the chair."
The Scottish government insisted it did not intent to abolish the position of rector at universities. A spokesman said: "The Scottish government recognises the contribution made by rectors.  Over the summer, we have been talking to all higher education stakeholders, including current rectors, about their views on how a model for elected chairs of court might work in practice.
"As the Higher Education Bill progresses through parliament, we will continue to listen to all constructive views and suggestions made."
The Higher Education Governance (Scotland) Bill would require governing bodies to include at least two members directly elected by staff, and two members each nominated by trade unions, students' unions and alumni associations.
The proposal that the chairs of the governing bodies be chosen through an election has proved controversial and led to claims that it could involve voters from outside the university.
Critics have also argued that trade union representatives could be torn between the interests of their members and the interests of the institution.
As well as a rector, Edinburgh University also has a vice convener who chairs several of the university's most important committees and helps hold the Principal to account.
Mr Morrison is a graduate of Edinburgh University and was the first student to run to become rector in the 1960s.
He later became the director of programmes and then chief executive of Granada Television. He left to found All3Media, Britain's largest independent television production group, and is still the company's non-executive chairman.
At his inauguration, attended by the Chancellor of the University, Princess Anne, Mr Morrison also called on students to use their time at university to equip themselves with the skills that will be required in the changing world of the digital economy.
The new rector of Edinburgh University has warned that government plans for a shake-up in how universities are governed could threaten the role.