Stefanini was said to have been rejected by the Vatican on the grounds of his alleged homosexuality.
Difficulties over Stefanini’s appointment have been reported by French and Italian media in recent days.
President Francois Hollande tipped him for the post in January, but so far the Vatican has failed to accept his credentials.
“Clearly, even at the Vatican they do not practise what they preach,’’ the chairman of the Arcigay association, Flavio Romani, said in a statement.
He accused “top prelates” of contradicting one of Pope Francis’ most famous remarks.
“If a person is gay and seeks God and has goodwill, who am I to judge?’’ the pontiff said in July 2013.
However, the Vatican has refused to comment on the affair.
It is unusual, but not unprecedented, for the Holy See to hold off on the acknowledgement of ambassadorial appointees, in the hope that they may be replaced by more palatable alternatives.
Stefanini, who was the second-ranking diplomat at the French embassy to the Holy See in 2001 to 2005, would not be the first gay ambassador posted to the Vatican.
Meanwhile, no diplomat in that position has ever been openly homosexual.
The polemic over his nomination comes in the wake of tensions between the Vatican and France’s Socialist government over the legalisation of gay marriage.
The law was enacted in 2013, to the outrage of Catholic and conservative groups.