1680x1050 is still widescreen resolution albeit 16:10 rather than 16:9. If you are using a decent software player such as any of the major ones like Arcsoft TotalMedia Theatre, Cyberlink PowerDVD, or Corel WinDVD, they will fit the content to your monitor. As long as the monitor is an HD monitor you will still see an amazing image but you just won't be seeing it at its full native resolution.
I use HDMI over DVI connecting my ATI Radeon HD 4870 videocard to my Samsung SynMaster 2232GW 22" monitor at 1680x1050 and I am thoroughly pleased with my HD-DVD/BD playback. At the time I purchased my 22" monitor the cost difference to bump up to a 24" monitor with 1920x1080 resolution was insanely prohibitive. Were I to buy a monitor now I'd go 24" since prices have fallen. My monitor is still in perfect condition and works beautifully so I have no intention of upgrading anytime soon.
The question of supporting "full HD" depends on your definition. Literally, only a 1920x1080 resolution is full HD resolution so only a monitor that is an HD monitor with that resolution or higher is "full HD". Having watched HD-DVD and BD playback on a Samsung monitor that was 24" (1920x1080) vs my 22" (1680x1050) monitor, I'm definitely not feeling depressed. Both look beautiful.