"Bolsonaro has shown all his misogyny with this veto," added Marilia Arraes, a left-wing legislator who was behind the bill.
"We cannot be silent, we're talking about the dignity of thousands of women."
She hopes to have the far-right leader's veto overturned in parliament.
"What century are we living in? Why do we have to fight for such obvious things? Once again us women have been disrespected. Menstrual poverty has been in our country for years," singer Preta Gil, the daughter of music icon Gilberto Gil, wrote on Instagram.
On Thursday night, Mr Bolsonaro said on his weekly Facebook speech that he would have to "manage" to find the money for the initiative if his veto is overturned.
The bill aimed to benefit five million women, notably students from poor neighbourhoods and prison inmates.
Mr Bolsonaro said the bill does not specify where the money would come from and that he would be forced to "take funds from the health or education budget" should it be passed.
"I'm not going to increase taxes or create a new one for this," he said.
According to the Girl Up NGO, created by the United Nations in 2010, a quarter of teenage girls have to miss several days of school a month due to "not being able to have their periods with dignity."
According to a UNICEF report, 713,000 Brazilian girls do not have toilets or showers in their homes and more than a quarter of a million do not have "access to necessary hygiene at school."