The 14-month-old coalition's senior member is the anti-establishment 5-Star Movement headed by rival Deputy Premier Luigi Di Maio, who has sparred with Salvini increasingly on key policy issues.
Conte has demanded Salvini, who is also interior minister, lay out his reasons in Parliament for refusing to no longer back the government he helped form after 2018 elections brought populists to power for the first time in Italy.
Salvini must "explain to the country and justify to the voters who believed in the prospect of change the reasons that lead him to interrupt ahead of time, abruptly, the work of the government," Conte said.
For a year now, the League's popularity has soared in opinion polls, local and regional balloting and the European Parliament elections earlier this year.
Those soundings have also shown a plunge in support for the oft-bickering 5-Star Movement, whose sometime left-leaning platform has kept the League from pulling the government further to the right.
"I don't look at the opinion polls," Salvini said Friday during an appearance in Termoli, a seaside town in the south. "I said 'stop' because when a government, instead of doing things, blocks them, it makes no more sense" to continue.
For months he has complained that the 5-Stars, often citing environmental concerns, were opposing plans involving highways, tunnels and other infrastructure projects. Salvini said earlier in the week his patience ended when the 5-Stars brought a parliamentary vote aimed at blocking construction of a high-speed rail link between France and Italy.
Rival Di Maio predicted Salvini's abrupt yanking of his support would backfire at the ballot box.
"When you take the country for a ride, sooner or later the bill arrives, sooner or later you pay the consequences," the Italian news agency ANSA quoted Di Maio as saying in a written statement.
League senators sought a confidence vote in Parliament on Conte's government that the premier appeared destined to lose, without Salvini's support.
When the League first joined a coalition government in 1994, it pulled the plug on what was then media mogul Silvio Berlusconi's first government by yanking out all its ministers.
The government is obliged to resign if it loses a confidence vote on it.
The Democrats, Italy's largest opposition party, are creeping up in opinion polls at the apparent expense of the 5-Stars, but they lag far behind the League.
Lawmakers have just started their summer vacation and it's unclear when they might be called back into session to hold debate and a vote.
Lawmakers "should move their rear ends and come to Parliament, if need be, even next week, because millions of Italians are also working the next week, and the week after," Salvini told a sea of cheering supporters Thursday night in Pescara, a small city on the Adriatic.
Although he himself has spent much of the last few weeks at the beach, Salvini frequently depicts himself as the champion of the working class.
Far-right Brothers of Italy leader Giorgia Meloni, a would-be ally in any League-led government, called on Salvini to cement a campaign alliance to bring both parties to power so they could enact what she called the "politically incorrect reforms" needed by the country.
An alliance with Meloni's staunchly nationalist party could see Italy's already often-tense relations with Brussels deteriorate.
Former center-left Premier Paolo Gentiloni tweeted a warning that Salvini was seeking "full powers to bring us outside of Europe."
Salvini regularly rails against EU budget rigor, insisting that more state spending on infrastructure will get Italy's long-stagnant economy growing again. His pro-Russia party also clashes with Brussels' sanctions against Moscow over its annexation of Crimea from Ukraine.
Italian prosecutors are investigating if a close Salvini associate had illegally schemed to obtain Russian funds for the League. Salvini contends he's never taken money from Russians.
Associated Press writer Colleen Barry in Milan contributed.