"Fretilin (the ruling party) has to choose," Gusmao said in a broadcast address to the nation that lasted nearly two hours.
"Ask Mari Alkatiri to take responsibility for the major crisis, about the sacrifices of the state, the law and democracy, or, tomorrow, I will send a letter to the parliament to inform them that I will resign from being president of the republic because I am ashamed of all the bad things that have happened."
Gusmao added that he "didn't have a brave face to show the people" after the violence in recent weeks.
Fierce gun battles between rival security forces in the capital last month raised fears that the country was plunging into a civil war.
At least 21 people were killed and about 150,000 residents fled from their homes, but the arrival of a 2,700-strong Australian-led peacekeeping mission several weeks ago helped to ease tensions.
Gusmao told Alkatiri, who has widely been criticised for not doing more to stop the violence, to resign or be sacked on Wednesday.
But Alkatiri said on Thursday that he would not resign and the ruling Fretilin party rallied behind him.
Alkatiri held emergency talks with members of his Fretilin party, which control 55 of 88 seats in the country's parliament, on Thursday, after which party spokesman Jose Reis said:
"Fretilin reaffirms Dr Mari Alkatiri as prime minister."
Gusmao immediately accused Fretilin of creating instability and demanded that the party replace Alkatiri as its leader within a week - though it was not clear what authority he has to make such a demand.
"If the situation does not change by early tomorrow morning, I will send an official letter of resignation to parliament," he said.
"Fretilin's leaders want to kill democracy in East Timor."