I do not have Flash installed on my computer.
Is there any way to watch videos on Facebook without using Flash?
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As of this writing Facebook supports HTML5 video but is still using Flash for Safari browsers. Why? Who knows. (source)
Works fine without Flash if you're using Firefox or Chrome. For now replacing the "www" with "m" in the video URL as described by Hjulle is the only workaround in Safari unless...
If you enable the Develop menu in Safari (Advanced Preferences), you can change your User Agent and see how Facebook skips Flash. (Fool Safari into playing videos without the Flash plugin).
Best option with Develop menu in Safari is to choose an iPad User Agent.
Yes, you can. Just go to the post* with the video and replace "www.facebook.com" in the url with "m.facebook.com". This will redirect to the mobile version of the page which has a direct link to the video.
*To get a link directly to the post, press the timestamp on the top of it.