If we just define fitness as pushing watts or burning calories, then no, there's no appreciable fitness benefit to riding fixed.

But it it is excellent training in two respects. It will teach you to keep pedaling. A lot of newbies pedal-pedal-pedal-coast. Riding fixed will break you of the pernicious coasting habit. It's almost always better to keep the pedals turning over even if you aren't applying pressure.

Even better, riding fixed will help you to broaden your power band. Because you only have one gear and can't coast, when going downhill you will be forced to spin out of your comfort zone. This teaches smoothness or supplesse. When going up or into the wind you will be forced to mash. This increases strength.

So there are real training benefits that I see.