Maduro said he was launching an "anti-coup commando squad" of hardline senior security officials.
Opposition majority lawmakers on Monday passed a motion declaring Maduro had effectively "abandoned his post" by failing to tackle the country's economic crisis.
"They will have to live with the consequences of their call for a coup d'etat yesterday in the National Assembly," he said in a televised address, referring to the opposition-controlled legislature.
Maduro said he was launching an "anti-coup commando squad" of hardline senior security officials. He did not specify what steps they might take against the opposition.
But about 300 Maduro loyalists took to the streets Tuesday to block lawmakers from entering a hospital where they had scheduled a public meeting to reach out to their own supporters.
The opposition lawmakers had to move their gathering to another hospital -- Caracas' main maternity facility.
"We are not going to stop getting out into the streets and be with the people who elected us," said Julio Borges, speaker of the National Assembly.
The opposition blames Maduro for an economic crisis that has prompted deadly riots and looting due to shortages of food and medicine.
Maduro says the crisis is the result of a US-backed capitalist conspiracy.
He defended his performance, listing a series of investments his government has made in public health, housing and education.
Maduro's supporters on Tuesday filed an appeal at the Supreme Court to overrule Monday's motion, said Hector Rodriguez, leader of the pro-government bloc in the assembly.
He said the appeal calls for criminal charges against the opposition leaders who proposed the declaration in the assembly.
The court has consistently backed Maduro in a series of rulings against the opposition since it took control of the assembly a year ago.
It had already said ahead of Monday's declaration that the assembly was not qualified to remove Maduro from office.