Fig tree issue

Before you start spraying, look at the base of the tree and see if there is any orange 'bark-flakes'. That would be a bug inside eating the wood. It will also happen on a single branch. Our fig had a single branch start will spots then all leaves fell off, we cut it off to the main trunk, and there was a bug inside the branch as large as my pinkie. But did lose an organic rose I loved to disease, sprayed in the end but too late to save it. Good luck.

There is a middle course between spray and no-spray, particularly when a fungus or insect is about to destroy a crop. The best place to consult is the Organic Materials Review Institute (https://www.omri.org/), which will tell you what is safe to use and what is not. Organic gardeners like me look for the OMRI label on fungicides and insecticides. It goes without saying that even an approved product is used sparingly.

Thanks so much everyone!

This is where taking in a sample of the leaves to a local nursery would give you immense help. Also your county agricultural extension is a wealth of knowledge. Check them out! They will have help that addresses your climate, etc. Good luck!

Enjoying reading this question also... Just planted a fig tree!

More information is needed to completely answer your question. I'll tell you what I know first, then tell you what I need to know.
What I know:
I avoid using any type of sprays on food products. I just never do it. Companies will always tell you how safe something is, but the point of growing your own is often times to eliminate the risks associated with chemical products. You can't fully trust chemical companies or even the USDA and FDA for that matter, companies give too much money in exchange for influence. It's just a fact of life. (read Marion Nestle's "Food Politics" to learn more).
That said, fungal and disease issues are extremely difficult to manage after they make an appearance. Preventive treatment before signs of infection is the only way to control/manage outbreaks. Your best option at this point would be to remove spotted leaves, avoid getting the foliage wet, and prevent the plant from obvious stress (excessive drying out, plenty of available nutrients, etc) be certain it receives plenty of light and maintains consistent soil moisture.
I'm guessing that the fruit will be fine to eat (obvious if it's not).
Now what would be helpful to know:
What is the fig variety or cultivar.
Is it in a pot or the ground.
Where are you and the plant located.
How long have you had it
How big is it
What exactly do the spots look like - all one color, symmetrical, dark center radiating lighter
All this information is helpful to make an accurate diagnosis

I have a healthy fig tree outside my front door. Are you sure the spots are caused by fungus and not some climate issue? Figs are very sensitive to location. You might ask at a nursery about eco-friendly treatments.