Viola spp are characteristic of base rich and poor fertility soils; therefore if you add some fertiliser to your lawn you will likely disfavour viola spp and also enhance the growth of vigorous grasses, ie you will provide conditions to favour grass growth and disfavour viola spp thereby getting rid of viola spp without recourse to herbicides
Which is roughly the opposite of what we (Warwickshire Wildlife Trust) do to encourage wild flowers on our grassland nature reserves. Reducing nutrient levels discourages the vigourous grasses. Wild meadow flowers don't compete successfully with vigorous grasses. There's more to it than that, but...
I've been gently encouraging dog violets in the lawn for several years. That's mostly about mowing around them when in flower, to let them set seed. As weeds, they're benign, like herb robert, but unlike the hawkweeds, our local wood avens, the wretched green alkanet and several others.
You evidently have a problem with your violets. Can you identify what sort (deliberately vague) they are?