The MP, Aidan Burley, has repeatedly apologised for the incidents during a drunken night out by a group of British men in the French Alpine ski resort of Val Thorens.
A preliminary investigation was launched in December after a complaint from a French anti-racism group, and local prosecutor Patrick Quincy told AFP that authorities were now moving forward with a full official investigation.
The investigation will focus on charges of defending war crimes or crimes against humanity, promoting racial hatred, wearing the uniform of an organisation that carried out crimes against humanity and making racist insults, Quincy said.
Burley, 33, was sacked from his job as a parliamentary private secretary to Transport Secretary Justine Greening after pictures and video taken at the stag party were published in the British press.
Burley, a lawmaker from Prime Minister David Cameron's centre-right Conservatives, represents Cannock Chase in Staffordshire, central England.
British stag parties, held before a man gets married, are typically jovial, boozy nights out, often with the groom-to-be in embarrassing fancy dress.
Burley was photographed sitting next to the stag, who was wearing the black uniform of a World War II-era German SS officer.
A video showed a guest raising a toast to the Third Reich at the party and reports said the group had later chanted "Mein Fuehrer!" and the names of Nazis Heinrich Himmler and Adolf Eichmann, who were responsible for the Holocaust.
The Conservative Party said in a statement that Burley was removed from his post because of his "offensive and foolish" behaviour at the party.
In a statement after the incident, Burley voiced his "deepest regret" and insisted: "I have no sympathies whatsoever with Nazism, racism, or fascism."
Under French law it is a crime to make anti-Semitic statements or exhibit
Nazi uniforms or emblems in public, unless required for a film, play or other cultural production.
Glorifying or defending war crimes or crimes against humanity is punishable by up to five years in prison and a fine of €45,000 ($59,000), while wearing the uniform of an organisation that carried out crimes against humanity is punishable by a fine of €1,500.
Burley was caught up in a fresh scandal last week when British media reported allegations that he had behaved disrespectfully during a trip to Auschwitz.
The reports, which were raised in the British House of Commons by an opposition Labour lawmaker, alleged that Burley had been sending text messages and dozing during a speech by a concentration camp survivor.
Sources close to Burley denied the allegations.