MOSCOW—Thousands of pro-Kremlin demonstrators and phalanxes of riot police took to the streets of downtown Moscow on Tuesday to squelch opposition protests, as Prime Minister Vladimir Putin played down the setback his United Russia party suffered in Sunday's parliamentary vote.

Antigovernment demonstrations were scattered and small, a day after nearly 10,000 people gathered in central Moscow—in one of the largest opposition protests in recent years—to denounce alleged rigging of the elections. Police detained hundreds of demonstrators....