Not long term. You can fertilize with products that aren't synthetic chemical fertilizer. Biosolids, manures, alfalfa pellets are a few options. Once you get a lawn weed free you can maintain it, if it's less than 1000 sq ft with mechanical methods (pulling weeds).
A lawn is basically a crop and you can grow it organically, just like any vegetable farm can do that. As a crop, other plants want to compete with it. Your job as a grass farmer is to do everything to help grass win the battle for space, soil, nutrients, water and sunlight.
Once grass is really winning the battle, you have less work and just need to help maintain dominance for those resources. Example :if you let nitrogen get to low conditions will favor plants that can source their own nitrogen, like clover. This is why clover can quickly come to dominate an under fertilized yard. Healthy soil with good nutrients and organic matter seems to allow for less fertilizer.