Parliament has to consider a debate on any petition posted on its website that attracts more than 100,000 signatures.
The proposal, posted before the June 23 referendum, said the government should hold another plebiscite on EU membership if the support for Leave or Remain in a referendum was less than 60 percent in a turnout of under 75 percent of eligible voters.
The result on Thursday saw 52 percent of voters, 17,410,742 people, back a British exit, on a turnout of 72 percent.
Since then, the petition — which only British citizens or UK residents have the right to sign — was proving so popular that by Saturday, 2,005,101 people had signed it.
By late afternoon, it appeared to be rising at a rate of more than 3,000 signatures a minute.
Most of those who signed were based in areas where support for staying in the EU was strongest, most especially London, the website indicated.
Prime Minister David Cameron, who said on Friday he would resign after leading the failed campaign to keep Britain in the EU, had said there would be no second referendum.