Vaultier told French TV he was a fan of Djokovic and had always wanted to meet the Serbian former world number one.
Djokovic responded swiftly on Twitter with an invitation to the French Open at Roland Garros.
“@pierre_vaultier let’s make it happen. See you at RG (Roland Garros)! And congratulations on your Olympic medal(s)!“
Vaultier said the likeness between himself and the tall, dark Djokovic had been raised in the past and added: “When I am watching him play, his expressions and his intonation, I feel like I am watching myself.”
The 30-year-old Vaultier won the Olympic snowboard cross gold in Pyeongchang on Thursday, retaining the title he won in Sochi four years earlier.
The French Open tennis championships will take place at Roland Garros from May 27 to June 10.