Facebook Inc., Twitter Inc. and Google Inc. have started resisting Russian government orders to remove information about a rally next month in support of opposition leader Alexei Navalny, raising the prospect of a showdown over the Kremlin’s efforts to control online information.

In response to a request from Russian prosecutors, Roskomnadzor, the country’s communications regulator, began issuing block orders for Russia just hours after the Moscow rally was publicized on social media late last week, officials said. Facebook...