Classes were suspended in federal universities and secondary schools in the first nationwide protests against the far-right president since he took office on January 1.
Protesters flocked onto the streets in Brazil's biggest cities, including Sao Paulo, Rio de Janeiro and Belo Horizonte, in marches that grew steadily larger as the day wore on.
The budget cuts feature a 30 percent reduction in subsidies paid to state-owned universities.
Bolsonaro blasted the protesters as being "useful idiots manipulated by an expert minority that makes up the heart of federal universities in Brazil."
He accused leftist militants of stoking the protests.
Speaking from Texas -- where he was to receive an award from the US-Brazil Chamber of Commerce -- Bolsonaro defended his education policy, saying he had inherited "a Brazil that was destroyed economically."
He added that necessary budget cuts had been "lower than forecast."
Some 15,000 protesters marched in the capital Brasilia, while 20,000 took part in demonstrations in the northern city of Belem, according to police.
Organizers said 70,000 people marched in the leftist stronghold of Salvador in the northeast.
Isolated incidents were reported. Police used tear gas to disperse protesters in the southern city of Porto Alegre.
"I have a master's degree and a post-doctorate in energy. If someone is an idiot in this story, it's not me," said Mariana Moura, a 38-year-old protester in Sao Paulo, angered at Bolsonaro's slur.
"Without science there is no health, or work. We are here fighting for Brazil to continue producing knowledge. Without money, there is no knowledge," said Moura, a researcher.
"Unfortunately, education is not a priority. That's why I'm here today to fight for public education," said Alessandra Roscoe, a writer, as she took part in the protest in Brasilia.
"Class today is in the street," read one banner in Rio. Demonstrators chanted: "Either stop the cuts or we stop Brazil."
In Sao Paulo, authorities closed Paulista Avenue, one of the economic capital's main arteries, which was crowded by mostly young protesters singing along to a band, and chanting the slogan "Books Yes, Weapons No."
Government 'not responsible
Education Minister Abraham Weintraub -- who sparked the protests by slashing the federal university subsidies -- was forced to defend his policy in Congress.
Weintraub insisted the government "is not responsible for the disaster of basic education in Brazil."
He warned that "university autonomy is not sovereignty. Universities have to respect the laws."
Since assuming the presidency, Bolsonaro has been on an ideological campaign to promote his ultraconservative ideas and values, and to erase any trace of the 2003-2016 leftist Workers' Party government -- including what he called "cultural Marxism" -- from classrooms.
Eduardo Bolsonaro, one of the president's three lawmaker sons, accused the Workers' Party of being behind Wednesday's massive street protests.
Several chief education officers claimed the budget cuts would compromise the ability of federal universities to operate and threatened to paralyze them.
A sudden suspension of financing for masters degrees and doctorate scholarships in the sciences and human sciences last week threw oil on the fire.
Although the cuts affect only federal institutions, the protest movement has been joined by many private universities such as Rio's Pontifical University which voted last week to join the nationwide demonstration.