From your description I doubt you have violets. It sounds to me like creeping Charlie. This low-growing, aggressive member of the mint family has roundish leaves and purple flowers in spring. It's a tough, invasive weed, especially in shady spots. You have three primary options, all of which have drawbacks. Hand-pulling. For large infestations, forget it! But if you spot it early, you can probably manage to eliminate creeping Charlie by hand.
Herbicide. Find a three-way weed killer (for example, one with Trimec in it) and use it a couple of times in spring or fall, when the soil is moist and growing conditions are good. The key is repeat applications a couple of weeks apart; one application alone will fail. Another option is glyphosate (Roundup). If applied directly to the plants' foliage, the nonselective herbicide does not go into the soil, so it should not affect your groundwater. It's designed to get inside the plant and kill it from the inside out. Glyphosate kills the grass, too, so you'll need to start new grass in treated areas.
Borax. If you apply it in the right amount, household borax is slightly more toxic to creeping Charlie than to grass, so you can kill the weed but spare the lawn. It does have its limits: It's going to burn the grass if you apply it too heavily, and you can apply it only once a year for 2 years before you exceed the level that will harm grass. The proper mixture is 10 ounces of borax in 21/2 gallons of water for 1,000 square feet of infested lawn.