Is there a natural way to reverse browning spots in lawns?

you need a soil test by your state Agricultural Agent to determine the CAUSE of the spots --it will depend on a variety of factors -- the kind of grass, the composition of the soil itself, water/shade/sunlight, and of course, whether or not any disease is affecting the lawn.

Just plant new grass seed in those spots and keep watered it will grow back green

Pour a glass of beer on the brown spots. It works like a charm. Beer has lots of fermented sugars and nutrients inside it and the dead patch will absorb them and feed the soil underneath. So the beer acts as a fertilizer and if the brown spots are caused by pets it will neutralize the nitrogen in their urine.

If the brown spots are caused by grubs (Japanese beetle larvae) then you need grub control. First determine if you have them. Dig up a square in late June and see if you see little white worms. Or in the summer, do you have Japanese beetles on your plants. If you have them, there is a natural product called milky spore that kills the larvae. It's expensive but it should take care of it for 5 years.