I was seriously injured at work when my work truck was hit by a car that ran a red light in D.C. I was delivering materials for my company. I'm getting workers comp benefits and the workers comp insurance company paid for my surgeries. Since the car accident was the other driver's fault, can I sue the other driver?

Yes.  You actually have 2 cases with different benefits or damages available.  The workers compensation insurance company should be paying for your time off work (66 2/3 % of your salary) and for all of your medical treatment - surgeries, hospitalizations, office visits, MRIs, physical therapy, etc. as you need it, even though the accident was caused by another driver.

You also have a case against the other driver (sometimes this is called a 3rd party case because the other driver isn't your employer or a co-worker) for the harms and losses he caused when he ran the red light and hit you.  These harms and losses can be the cost of your medical treatment (both now and in the future), lost income (again, now and in the future), pain, suffering, permanent injury, disfigurment and other things that have been taken from you.

And you can pursue both cases.  The catch (there always is one) is that the workers comp insurance company gets paid back what it paid to you or on your behalf for medical treatment.

There are a couple of traps to watch out for.  For example, check out the question about time limits for filing a 3rd party case in D.C.