The Russian watchdog said that if Google failed to respond to its request, it would consider it "interference in its sovereign affairs" and "hostile influence [over] and obstruction of democratic elections in Russia".
"A number of organizations owning YouTube channels acquire advertising tools from YouTube, including push notifications, in order to disseminate information about unauthorized (illegal) mass events, including those aimed at disrupting federal and regional elections", Roskomnadzor said in a statement.
The sanctioned demonstration on Saturday attracted an estimated crowd of more than 50,000 people, the largest turnout at a Moscow protest in several years. The rally on August 10 was the fourth in support of independent candidates at the upcoming election to the Moscow City Duma (parliament).
Of additional concern to the government is that some of Russia's "most famous internet celebrities and musicians" have urged their millions of followers to join the demonstrators, according to the Wall Street Journal.
Over 300 anti-Putin demonstrators were also arrested throughout the day.
It wasn't the first time Google's come under pressure from Roscomnadzor - it reportedly tweaked its search results in Russian Federation to remove blacklisted websites earlier this year, after being threatened with fines if it failed to do so.
A law enacted this year requires the routing of internet traffic through Russian servers, a move that critics claim could lead to a firewall similar to a system in China that blocks political content and prevents Chinese users from using sites such as Facebook, Twitter and Google.
Roskomnadzor set out its requirements to Google in the form of a letter. In contrast to previous protests, the authorities had approved this rally.