The law has stoked concerns that Hindu nationalist Prime Minister Narendra Modi's government wants to marginalise India's Muslim minority.
The short video clip shows two bearded men in traditional Muslim clothing discussing the legislation before concluding that the country can only progress if there is "peace and brotherhood".
India protests rage on as death toll rises to 25
The video, which was released by Modi's Bharatiya Janata Party on its Twitter handle, has garnered "likes" and also ridicule.
The ruling right-wing government also carried an advertisement across all national dailies, with a "myths vs facts" explainer to support its claim that the law was not against India's 200 million Muslims.
The advert claimed there were no immediate plans to roll out a nationwide register of citizens, which has stoked fears of Muslims and others unable to prove they are Indian becoming stateless.
Even if the register would be rolled out, "the guidelines would be framed such that no Indian citizen would face any harassment whatsoever", the advertisement read.
The wave of protests across the country marks the biggest challenge to Modi's government since sweeping to power in the world's largest democracy in 2014.
At least 25 people have died in the demonstrations over a span of two weeks.
The citizenship law, passed by parliament earlier this month, allows people of six religions from Muslim-majority Pakistan, Bangladesh and Afghanistan a path to citizenship.
Muslims, however, are excluded from the group of six.
Islamic groups, the opposition and others at home and abroad fear this forms part of Modi's aim to remould the country as a Hindu nation.
Demonstrations have rocked India over the past few weeks, while heavy-handed police tactics including the storming of a Delhi university a week ago have fuelled anger.
Indian authorities have cut mobile internet access in places and imposed emergency laws banning assemblies.
In a speech on Sunday, Modi sought to reassure the Muslim community saying they "don't need to worry at all" and laying emphasis on India's diversity.
In the coming few days, his party says it will contact more than 30 million families, organise countrywide rallies and hold over 250 press conferences to dispel doubts over the new law.
Hundreds arrested in India during days of protests over citizenship law
The protests that first started in the northeastern state of Assam have since spread to the rest of the country, with Uttar Pradesh -- a tinderbox state with a population of over 200 million -- seeing the maximum deaths.
More than 20 people have died, mostly from firearm injuries, in the state including a schoolboy who was crushed during a stampede that broke out as police tried to disperse the protesters.
Fresh protests were planned on Monday in New Delhi as well as in Kochi in the south.
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