MOSCOW—The Russian ruble fell to fresh record lows Thursday as investors sold the currency heavily a day after the central bank said it would cut back costly intervention and allow the exchange rate to be set mainly by market forces.

“The next few days will be crucial for the ruble as the market will put to the test the central bank,” said Piotr Matys, emerging market strategist at Rabobank. The ruble dropped steadily through much of the day Thursday, breaking through 45 per dollar to close at 46.56, marking a 30% decline...