The high-speed hydrofoil ship was en route to Sado Island from the port of Niigata on Saturday.
Ferry operators said the vessel hit an object, leaving a 15cm (6in) crack in its stern.
Public broadcaster NHK quoted a marine expert who said the scale of the impact suggests the ship struck a whale.
In a statement, the ferry operator apologised to its customers and said the ship looked to have collided with “marine life”.
“My throat hit the seat in front of me,” one passenger told local media. “People around me were moaning [in pain].”
There were 121 passengers and four crew members on board at the time.
Sado Steam Ship Company, which operates the Ginga ferry, says it managed to reach its destination under its own power about an hour behind schedule.
The Coast Guard reportedly said 13 were in a serious condition, although they were conscious. The injured have been taken to hospital.
The boat is powered by jets of seawater and can travel at up to 80km/h (49mph). One of the hydrofoil wings on the ship was also damaged in the collision.
Minke and humpback whales are reportedly migrating through the Sea of Japan at this time of year.