While I'm not sure of the protocols involved or the actual digital bitstream used, the data on the disc most definitely is completely read by the DVD player first.
It is digitally read, then goes through the DVD players processing and information is converted to 480i, 480p, 720p, or 1080i (depending on player) and then is sent via HDMI digitally to the display. At no point prior to the display device is the video signal converted out of the digital realm. Though, processing most definitely does occur to the video.