Sparring politicians in the recession-ravaged country have been at loggerheads for two months ever since elections in February that left no clear winner and revealed growing social unease.
The main parties appealed to the 87-year-old on Saturday to stay on - the first Italian president to do so - and elected him later in the same day despite his earlier insistence he would retire.
Reform efforts "have petered out thanks to the deafness of the same political forces that asked me to stay on", Napolitano said in a harsh speech in which he was sometimes overcome with emotion.
"If I am faced with this deafness again, I will not hesitate to draw my conclusions," he told lawmakers in an apparent hint that he could resign or call new elections if the deadlock continues.
"A government must be formed without delay," he said, stressing that the country was in a "fatal stall" and that failed votes to elect a president last week were a source of "dramatic alarm".
He said the impasse had shown up "mistakes, omissions and irresponsibility" by politicians.
Napolitano's re-election was slammed by the new anti-establishment Five Star Movement party, which has called for sweeping changes to Italy's traditional political system, as a "mini-coup".
But the financial markets cheered the news.
Stocks in Milan closed 1.66 percent higher and Italy's borrowing costs on ten-year government bonds fell to 4.059 percent - their lowest level since 2010, indicating investor confidence.
Napolitano is an ex-communist who began his political career as an anti-fascist activist during World War II and is widely respected for his neutrality and gravitas.
Political observers say his nominee for prime minister could be announced on Wednesday or Thursday and the name most frequently mentioned is that of former premier Giuliano Amato.
Amato, 74, is seen as being above the party political fray but his nomination would fail to answer growing calls for wholesale change in Italy, especially from younger generations.
Amato's first government dates back to the early 1990s after a series of anti-corruption inquiries brought down the country's ruling class.
Nicknamed "Dr Subtle" for his ability to negotiate treacherous political waters, Amato was prime minister again in 2000-2001.
Analysts say the government could resemble the outgoing one of Mario Monti, a former European commissioner installed in 2011 after then premier Silvio Berlusconi was ousted by a rebellion within his own ranks and a collapse of financial markets.
The difference from Monti's "technocratic" government could be the inclusion of party political figures as well, analysts said.
The centre-left came first in February's election but failed to win an overall majority in parliament with Berlusconi's coalition a very close second and former comedian turned firebrand Beppe Grillo's protest movement in third place.
The centre-left has so far excluded an alliance with Berlusconi's centre-right but that may change.
Napolitano on Monday urged all political parties to be realistic about the composition of the parliament and said there had to be some kind of cross-party compromise to form a government.
"The aversion to an alliance is a sign of regression," he said, pointing out that most European governments were multi-party coalitions.
Napolitano said he would stay in office "until the situation in the country and in the institutions suggest otherwise and until my strengths allow".
Analysts have said Napolitano could step down within months after a new government is formed and will not serve out his seven-year mandate.
The centre-left has been riven by divisions since the elections and the party's leadership resigned en masse on Friday after two presidential candidates it had proposed failed to get elected.
Grillo's movement has capitalised on a growing sense that mainstream politicians are incapable of dealing with this social fallout.
The Corriere della Sera daily said in an editorial: "The holiday from reality is over."
"The new government, whoever it is led by, will have to give a rapid and concrete response to the problems of the country or Grillo will win."