He said Greece, unable to service its debt, would instead seek a bridge loan.
He told parliament he would keep all pre-election pledges, promising to raise the minimum wage, pay a pension bonus and rehire public workers.
Mr Tsipras's far-left Syriza party won elections last month on a promise to end austerity measures.
EU officials have rejected his efforts to renegotiate Greece's bailout terms.
"The bailout failed," Mr Tsipras said on Sunday, in his first major speech to parliament since becoming prime minister. "The new government is not justified in asking for an extension... because it cannot ask for an extension of mistakes."
"After five years of bailout barbarity, our people cannot take any more."
'Humanitarian crisis'
In last month's elections, Syriza fell just short of an outright majority and formed a coalition government with the right-wing Independent Greeks.
On Sunday, Mr Tsipras said the government's "irreversible decision is to implement in full our pre-elections pledges".
The first priority, he said, was "tackling the big wounds of the bailout, tackling the humanitarian crisis".
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That included giving free food and electricity to those worst affected by the economic crisis and ending an unpopular annual levy on private property.
Among other commitments outlined on Sunday were:
• a gradual rise in the minimum wage to €751 (£557; $850) by 2016
• payment of a bonus to low-income pensioners
• reinstatement of public sector employees "fired illegally"
• the creation of a new national broadcaster
Two of the measures - raising the minimum wage and restoring a tax-free threshold to €12,000 - contravene reforms made previously as conditions for receiving bailout money.
Mr Tsipras also announced a number of measures aimed at cutting costs or raising revenue, including
• a new tax on large properties
• a special portfolio to oversee fight against corruption and tax evasion
• a pension fund using revenues from natural resources
• cutting ministry cars and government aeroplanes.
The Greek prime minister also repeated demands that Germany - Greece's biggest creditor - pay reparations for World War Two and repay a loan that the Nazis forced the Bank of Greece to pay when they occupied Greece.
Greece had "a moral obligation to our people, to history, to all European peoples who fought and gave their blood against Nazism", he said.
Diplomatic tour
Greece's current programme of loans ends on 28 February. A final €7.2bn is still to be negotiated, but Greece wants permission to issue additional short-term debt while it seeks a new deal.
Mr Tsipras said Greece wanted to service its debt. "If our peers want so, too, they are invited to come to the table of dialogue so we can discuss how to make it viable," he added.
Mr Tsipras and his Finance Minister, Yanis Varoufakis, went on a diplomatic tour this week to try to reassure eurozone leaders about their plans.
However, Jeroen Dijsselbloem, who chairs the Eurogroup made up of eurozone finance ministers, said on Friday that Greece had to apply for a bailout extension if it wanted continued backing from the eurozone. "We don't do bridging loans," he said.
The European Central Bank has also issued a statement saying Greek banks could no longer access ECB credit by using Greek government bonds or bonds guaranteed by the government.
Greek debt stands at more than €320bn, or about 174% of Greece's economic output.
Eurozone finance ministers are due to meet on Wednesday to discuss Greece's debt proposals.
Mr Varoufakis told Italian television on Sunday that the euro was as "fragile" as a house of cards. "If you take out the Greek card the others will collapse," he said.
Source: BBC