This is a bit like asking for the best way to be a carnivore, without wanting to kill anything.
Grass, as it exists on lawns, cut and manicured, and frankly underutilized as anything other than ornamentation, is just not environmentally friendly. you can't run from that fact.
As you mow it, as you fill your mower with gas, as you put down chemicals which kill the soil, you are doing damage. Full stop.
If you still want to know, recognizing that you are doing damage, how to do as little damage as possible. then consider what your soil needs. (focus on healthy soils, not the stuff growing on top.) your soil needs organic matter, lots of it. no chemicals. chemicals will kill all the stuff grown inside the soil (or create an environment where living things in the soil are disadvantaged). and soil needs to be left alone largely.
So, assuming you can't be convinced to get rid of the lawn and do a garden, just sprinkle some compost over the top every so often, if loosely put on top then the grass will grow up through it just fine. don't cut the grass very short, let it grow taller than normal and cut it to a higher height, which will allow roots to grow deep, adding more living organic material down there. that organic material is really all the fertilizer you need. Once in spring and once in fall is probably all that is needed, but that does depend on what kind of soil you are starting with, sandy or clay.