Twenty-three other crew members on the USS San Francisco were being treated "for a range of injuries including broken bones, lacerations, bruises and a back injury", following the accident on Friday, Petty Officer Alyssa Batarla said.
The accident occurred as the vessel was conducting underwater operations 560km south of its base in Guam, the navy said.
The submarine was due to arrive back in port on Monday. An investigation into the accident has already started, officials said.
The navy said the submarine's nuclear plant had not been damaged and its hull was still intact.The submarine was heading back to base under its own steam, but on the surface.
The grounding occurred as the Los Angeles class submarine with its crew of 137 was heading for a port visit in Brisbane, Australia.
"There were no reports of damage to the reactor plant, which is operating normally," the navy said in a statement on Saturday.
Los Angeles class submarines are 110 metres long and have one nuclear reactor and one shaft.
The USS San Francisco is one of three submarines of its class to be based in Guam, where it has been there since 2002.
It can carry out intelligence gathering and take special forces on missions. Its strike arms usually include Tomahawk missiles.