Sir Richard Gozney will be paid around £100,000 a year as the Queen's latest representative on the island and live in Government House.
A Celtic League spokesman said using public money to fund the position "was outrageous at a time of austerity".
A government spokesman said the full cost of the role is £400,000 a year including all associated salaries.
Mr Gozney, a former governor of Bermuda, became the Isle of Man's 30th Lieutenant Governor when he was sworn in on 27 May.
He succeeds Adam Wood who stepped down in April after five years.
Bernard Moffatt, of the Celtic League said the position was a "relic of the colonial past" and "could not be justified."
Sir Richard represented the Isle of Man in Orkney at a commemoration to those who lost their lives at the Battle of Jutland, a key naval engagement of World War One.
He said: "Because of the Isle of Man's strong and heroic maritime and naval history, Government and I thought it right to accept the invitation."
The Lieutenant Governor acts as the representative on the Isle of Man of the Lord of Mann, currently Queen Elizabeth II.
No Manx-born person has ever been appointed to the role.
The "huge costs" of having a Lieutenant Governor for the Isle of Man "cannot be justified," a nationalist group said.