The "March for Science", held in 27 cities, also urged the government to allocate at least 3 per cent of the country's GDP to scientific and technological research.
The RSS-linked science organisation Vijnana Bharati dubbed the march "politically motivated, left-oriented and "anti-government".
In Delhi, some 400-500 scientists, researchers, students and others took part in the rally from Mandi House to Jantar Mantar.
Vinay Kumar, a mathematics professor at the Zakir Hussain College here and a part of the Delhi Organising Committee, said a memorandum with a set of demands had been submitted to the Prime Minister's Office.
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It urged the government to "ensure that the education system imparts only ideas that are supported by scientific evidence and enact policies based on evidence-based science".
In Mumbai, some 250 scientists, educationists and others participated in the march from Wilson College to Azad Maidan.
Aniket Sule, an astrophysicist by training and educationist with the Mumbai-based Homi Bhabha Centre for Science, denounced the "propagation of pseudo-science under the pretext of research" and expressed concern at the government's emphasis on Panchgavya, a concoction of cow dung, cow urine, milk, curd and ghee.
"We have demanded that this pseduo-science be stopped under the name of research on Panchgavya and cow urine. Science should be evidence-based," Sule told PTI.
Vijnana Bharati secretary general A Jaykumar described the protest march as "shocking" and said "renowned intellectuals" were not only "diminishing the reputation of the government, but that of the scientific fraternity".
He said the budgetary allocations for the Ministry of Science and Technology and Earth Science had not been reduced.
India spent about 0.88 per cent of its GDP on research and development in 2011-12, compared to US's 2.79 per cent and South Korea's 3.36 per cent.
The march -- held in Bengaluru, Mysore, Pune, Lucknow and other cities -- was initiated by breakthrough-india.Org, a science portal, Sule said. But the cities had their separate chapters which organised the protests, he said.
When contacted, IGIB Director Sanjay Kumar said this was done because of security concerns.
"Some scientists wrote to me asking for my permission to participate in the march. How will I allow them to participate when there is a mob? It was done due to security concern," Kumar said.
On April 22, 2017, more than one million people in over 600 cities around the world participated in a similar protest.