Thank you. I knew there was a JS solution, but was trying to avoid it if possible. Now that I've spent days trying to get this working without JS, I'm ready to go this route.
I just installed the "FitVids for WordPress" plugin, which uses jquery.fitvids.js. It does the trick for YouTube videos. In the FitVids option panel I gave it a class name (for testing, I used .brian), and then in my text widget I pasted the embed code and wrapped it with <div class="brian"></div>.
However, this solution doesn't work for the video that I need to embed (which uses "object" and "embed" rather than "iframe"). I checked the jquery.fitvids.js code, and it includes the "object" and "embed" selectors, so I'm still not sure what the problem is.