Yes, the moon does rotate. In a tidally locked system, the angular speed of the rotation of the satellite about its own axis is equal to the angular speed of the rotation of the satellite about its primary.
Thus, the moon has an angular speed about the Earth of 1 cycle / 28 days, and an angular speed of 1 rotation / 28 days about its own axis; this is why we only see one face of the moon.
The moon itself moves along a geodesic, however the individual particles that make up the moon as they rotate about the moon's axis of rotation do not.
If you took away the Earth magically, the geodesic the moon would subsequently take through spacetime would resemble a straight line. However, absent some external torque that slows the moon down, the pieces of the moon would continue rotating around the moon's axis; the centripetal force that maintained their circular motion was not the Earth's gravity, but the Moon's gravity, which is still present.