Sport photography question

Is a 70-200 f/4L sufficient for indoor and nighttime sport photography when combined with a speedlite 580EX?

It will work but, you need to bump up the ISO to get a decent shot of fast moving action.

It won't work well in low light. It's too slow, and a lot of indoor sports venues won't let you use flash.

No, not if you are looking for many usable images. At 200mm, you'd have to be blasting that flash, which would upset a whole lotta people most likely. You can, in many places, set up strobes and bounce them, though.

There's few things to consider. First for most of sports you need times around 1/500 or faster... or at least not much under 1/500 (unless you are going for some artistic photos). With f4 and even on longer side of lens you will have a lot of problems getting those times even at iso 1600, most of time (at least in school gyms) even at iso 3200.
Flash doesn't really bother athletes, since they never look at you (or at least they are not suppose to ), so they don't even see flashes. But with (usually crapy) indoor light you have another color temperature to deal with, not to mention all the shadows you get.
So it will be possible, but definitely not best way to work.

And don't forget that gym lights cycle, so the lighting is often inconsistent.

it will be hell. besides all the items listed above, your camera needs light to focus. The flash does little to non to help with this issue. If it is able to focus, it will be slow and have a hard time tracking. If you could setup for a particular location like right under the hoop for Basketball, a spot on some equipment for gymnastics... and you can prefocus... it might work. But really, you are seriously handicaping your ability to track subjects.

Sorry

My wife just took some shots at her sister's highschool graduation last week with our 70-210 f/3.5-4.5. The ceremony was in a local arena where sports and concerts are held. She had the ISO cranked to 1600 and in Av mode all she could get was an average shutter speed of 1/160. Since the f/4 version is only 1/3 stop faster wide open at 200mm that is about 1/200. Definately not fast enough for sports and possibly some hand shake as well.