In a series of tweets published through his official account, the on-loan QPR player questioned whether Brazilian defender Silva was "pre-op or post-op" and demanded an explanation why the 28-year-old had mentioned him in his pre-match press conference ahead of Tuesday night's Champions League quarter-final with Barcelona.
The Ligue 1 leaders expressed their unhappiness with Barton's "unacceptable remarks" and threatened to "take action" against the former England international.
A statement published on their website read: "In the last few hours, on social networks, unacceptable remarks have been made by Marseille player Joey Barton against Paris St Germain player Thiago Silva.
"These attacks are serious and go beyond mere verbal jousting.
"The directors of the club, the players and the coaching staff are supportive of Thiago Silva and cannot accept such statements.
"They reaffirm their unwavering commitment to the values of respect that should govern our sport. Thiago Silva and Paris St Germain reserve the right to take any action they deem necessary."
PSG have taken issue with a series of tweets against the player which began to appear on Barton's account from Tuesday afternoon. Barton originally tweeted: "Thiago Silva. That the same p**** thats been injured all season. Another over rated Brazilian. Sort your hamstrings out FatBoy..."
The Liverpudlian was responding negatively to comments from Silva ahead of the game at the Parc des Princes, which finished 2-2 ahead of next week's second leg at the Nou Camp.
He continued tweeting during the game, adding: "Have to take back what I said about Thiago Silva being over rated today. Been immense tonight. Still looks like a overweight ladyboy though!"
Marseille said in a statement that the club and Barton were keen to apologise for the midfielder's "inappropriate comments", adding: "The management asked Joey Barton to put an end to the attack on PSG's Brazilian defender. The Marseille midfielder pledged to immediately cease the posting of anything derogatory about Thiago Silva."
Barton has been summoned by the French football federation's Ethics Committee over the incident.
"Joey Barton has been summoned by the Ethics Committee and will be heard on April 15," committee president Laurent Davenas told Reuters.
The anti-homophobia group Paris Foot Gay said they had discussed Barton's comments with French league chiefs, who initially did not intend to take any action because the incident did not happen on a football pitch.
Paris Foot Gay said that Barton's comments were "without doubt based on the stereotype linking Brazil to transsexualism".
In a statement on their website, they added: "Paris Foot Gay considers this type of remark to be vulgar, discriminatory and funny to nobody and it should not be tolerated in someone with such a profile. To be a football idol excuses nothing and should, on the contrary, make players responsible for the impact of their remarks on the public (youngsters) especially."
Barton's tweet caused a stir in French media, making the front page of sports daily L'Equipe which asked: "That is British humour?"
Silva has not responded on Twitter.
Barton arrived in France on loan from Queens Park Rangers following a colourful career in England during which he spent two-and-a-half months in prison for assault during a night out in Liverpool.
At the time of the attack, he was already on bail for attacking former Manchester City team mate Ousmane Dabo and he stubbed out a cigar in the eye of a youth-team player at the club in 2004.