Wiggins had performed well in the time trials on the Tour
Moreni, who was Italian champion in 2004, tested positive for testosterone following last Thursday's 11th stage.
Wiggins, the reigning Olympic pursuit champion, was lying 138th - two hours 42 minutes and 20 seconds off the lead.
"I don't want to continue in the Tour de France anyway, it is not supposed to be like this," he told the Guardian.
"It is completely gutting to have to quit the Tour but everyone knows where I stand on doping. I have nothing to hide.
"It's pure stupidity on the part of Moreni. I don't know how he can have slipped through the net.
"It makes you think about your future as a professional cyclist. It makes me question the whole thing, but then you think why not continue because I get a lot of pleasure out of it."
Cofidis team president Francois Migraine said: "It's the only thing to do in such circumstances."
Wiggins finished fourth in the prologue, which was held in London, and fifth in last Saturday's 54km time-trial in Albi.
The 27-year-old would have expected to do well in next Saturday's time-trial between Cognac and Angouleme.
606: COMMENT The best start ever to a Tour de France - that is what the organisers were saying in London 18 days ago; it seems like a long time ago now
BBC Sport's Phil Sheehan
Wiggins' trainer Shane Sutton told Five Live: "Bradley has always been clean from day one and he will be very disappointed.
"One mindless individual has put everybody at risk and it is a shame."
London's mayor Ken Livingstone, who watched Wiggins in the prologue, said: "This is deeply disappointing for Bradley Wiggins, who has had a fantastic Tour de France.
"As a steadfastly clean rider, he is well-known for his vocal condemnation of drug use in his sport, and has been a tremendous ambassador for London and cycling."
Cofidis are one of 11 teams not to win a stage yet.
Their voluntary withdrawal leaves 19 teams in the race after Astana were asked to withdraw following Alexandre Vinokourov's positive test.
Vinokourov, a pre-race favourite, tested positive for blood doping after winning last Saturday's time-trial stage.
The Kazakh rider reportedly had two different kinds of red blood cells, indicating he has taken blood from someone of a compatible group.
Just 24 hours later, Moreni became the second competitor to fail a drugs test.
The Italian was questioned by police and Cofidis team manager Eric Boyer told Reuters: "He accepted his wrongdoing and did not ask for a B-sample."
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Stage 16 profile
Michael Rasmussen of Denmark has won stage 16 of the 2007 Tour de France in a time of 6h 23' 21".

Rasmussen increased his overall lead by 35" over Alberto Contador of Spain, who finished third.

After a rest day, the riders faced the third-longest, 218.5 km, and arguably the Tour’s hardest, stage. The first challenge was the brutal Port de Larrau (14.7 km climb at 8.1%), which is just 79km into the stage. The race dips into Spain for some 50 kilometres at the summit, the fourth country visited in this year’s Tour. From there, there was little reprieve for the riders, hitting the Alto Laza and the Col de la Pierre St Martin (14.2 km climb at 5.2%) before descending to the start of the difficult, and very steep, first-category Col de Marie-Blanque (9.3 km climb at 7.4%). The riders then climb the infamous Col d'Aubisque (16.7 km climb at 7%), which will reward out-and-out climbing ability and punish the legs of all the major contenders.

The entire Astana Team withdrew from the Tour prior to this stage due to Alexander Vinokourov testing positive for blood doping. Some riders staged a protest against doping at the start.

Another doping scandal hit the Tour today. Cristian Moreni of Italy tested positive for testosterone in a test conducted after Thursday's stage 11. Moreni was questioned by police. Cofidis team manager Eric Boyer told ''Reuters'': "He accepted his wrongdoing and did not ask for a B-sample."

Like Astana did, Cofidis has withdrawn its entire team from the Tour. François Migraine, the team president and CEO of the sponsoring insurance company, Cofidis, said: "It's the only thing to do in such circumstances."