yes you can see the youtube vid in full screen if the video was uploaded with HQ mode. Alot of video on youtube nowadays are in HQ mode.
If the vid is HQ, just select the video, press Menu > More > Watch in High Quality. That will set the full screen mode, trust me the color is amazing.
Remember the droid screen is 800 x 480 pixels. If the vid was uploaded in standard mode, you will see the smaller video resolution.