An RCD can include a housing assembly containing a bearing assembly and a rotating seal which seals an annulus between a tubular string and a body of the RCD, and a remotely operable clamp device which selectively permits and prevents displacement of the housing assembly relative to the body. A method of remotely operating a clamp device can include rotating a threaded member which is secured relative to an RCD body, and displacing a clamp section in response to the threaded member rotation, thereby selectively securing and releasing a bearing assembly and an annular seal relative to the body. Another RCD can include a rotating seal which seals an annulus between a tubular string and a body of the RCD, and a remotely operable clamp device which selectively permits and prevents access to an interior of the body, the clamp device including a motor which rotates a threaded member.