In a speech, he said that he had not assumed power "under the ignominy of a coup d'etat".
The army had complied with the constitution, he said, and he had reached the presidency "as the result of an absolutely legal transition process".
Congress said he would serve until 27 January, when Mr Zelaya's term was due to expire. Presidential elections are planned for 29 November and Mr Micheletti promised these would go ahead.
Both Congress and the courts had opposed Mr Zelaya's referendum, which asked Hondurans to endorse a vote on unspecified constitutional changes alongside the November elections.
Tensions over the issue had been escalating for several days, with the army refusing to help with preparations for the referendum.
Just before dawn on Sunday, troops stormed the president's residence. There was confusion over his whereabouts for several hours before he turned up in Costa Rica.
Mr Zelaya called his ouster "a plot by a very voracious elite, an elite which wants only to keep this country isolated, in an extreme level of poverty".
Troops arrest Honduran president
He urged Hondurans to resist those who had removed him and late on Sunday flew to Nicaragua for a meeting of regional leaders.
Congress said it had voted to remove him because of his "repeated violations of the constitution and the law and disregard of orders and judgments of the institutions".
One man told the BBC that he had been in the city's main square all day, along with 2,000 Zelaya supporters. Jeronimo Pastor described the situation as tense and called on the international community to get involved.
But another resident of the capital said people were relieved at Mr Zelaya's removal. "Now we have a new president and will have elections and things will go back to normal," Kenneth Bustillo told the BBC.
The removal of Mr Zelaya has drawn criticism across Latin America and the wide world.
The Organization of American States held an emergency meeting, while UN Secretary General Ban Ki-moon called for "the reinstatement of the democratically elected representatives of the country".
US President Barack Obama urged Honduras to "respect the rule of law" and a State Department official said America recognised Mr Zelaya as the duly elected president. The European Union called for "a swift return to constitutional normality".
Venezuelan President Hugo Chavez, meanwhile, blamed "the Yankee empire", and threatened military action should the Venezuelan ambassador to Honduras be attacked.
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