You could kill it by suffocating it from light and heating it naturally.
Put a dark tarp over it, and anchor it so it doesn't blow away.
The heat from the sun and lack of light will burn that and anything else growing under there in several days to a couple of weeks.
gardening.stackexchange.com/users/48/trip0d199
You need to keep your grass healthy to help combat ground ivy.
Water deeply and infrequently and fertilize properly.
Have a soil test done at your local cooperative extension to see if there are any issues you need to correct and how much fertilizer you need to add.
Ground ivy thrives in damp and insufficiently fertilized soils.
There have been university studies done on using Borax to kill ground ivy with some positive results but some tests didn't show any improvement.
On my blog, I show how the best way I found to remove ground ivy is with a thatch rake.
In the fall I use a manual thatch rake to dethatch my lawn.
If necessary I follow up by overseeding with a good grass seed and spreading about 1/3" layer of compost on top then water as directed for the seed.
The way ground ivy grows and roots makes this very effective.
It pulls up some other weeds too but not your grass as long as it's healthy.
First time I tried it
I was in shock how well it worked.
gardening.stackexchange.com/users/2448/OrganicLawnDIY
I severely disabled some ground ivy by letting it grow a foot tall and mowing it off at one inch.
The grass really jumped and thickened, though it was a shock at first.
The grass thickening crowded out the ivy and the ivy has been stunted all year.
gardening.stackexchange.com/users/419/J. Musser
You can kill ivy with storm windows/glass.
I got rid of a big patch after all else failed with several 4'x6' storm windows.
I laid the glass out on the ivy and it fried it in a couple hours.
I moved the windows around for a week and killed a 30 foot x 45 foot area of ivy.
Cook it and kill it.
gardening.stackexchange.com/users/2472/don woods
I've effectively killed it with a 20% horticultural vinegar using a pump sprayer.
The grass around it will look like it died but actually goes dormant like it does in a drought.
If you water it the day after you spray the vinegar, it will turn green.
With the ground ivy, you have to be vigilant because if you leave one runner...
the problem comes back.
A week after a spray, I'll look for more and do it again.
Also, look to see from where it's coming.
If your neighbors have it and don't do anything about it, the problem will be endless.
I recommend attacking the problem together if possible.
Raising the pH with lime, spraying with a bacterial based compost tea and adding calcium and magnesium will help.
Sweetening your soil will also help keep out dandelions.
gardening.stackexchange.com/users/2887/Suzanne at Organic Garden Co
You could just hire someone else to painstakingly pull it all out, like maybe a neighborhood kid in need of odd jobs.
That might be cheaper and less effort than finding the perfect way to kill it.
Plus, being social and offering jobs can be a good thing.
Remember, a tool to help pull weeds goes a long way, even if it is just a regular knife.
With a tool, the plant doesn't break off at the roots, generally.
Just stick it in the ground by the roots and pull back to loosen them; then pull it up.
Granted, I haven't tried using a tool with creeping charlie (we don't have that around here, although we have something that looks exactly like it), but it worked great for our weeds this summer
(I think we mostly had bind weed or something similar).
Anyway, tools make weed pulling more enjoyable for me.
Too bad I didn't know about that trick as a child.
We had monster weeds, then, and grasshoppers to match.
gardening.stackexchange.com/users/7111/Brōtsyorfuzthrāx
I agree with the above comment about the structure of your lawn - I live in New Hampshire and typically have
acidic soil - creeping Charlie likes damp acidic soil - typically in mulched areas, and it spreads to the lawn area.
I've been putting down heavy amounts of lime to raise the alkaline level, but will try thatching this fall and borax in the spring!!
(it takes a bit if time for the lime to break down into the soil).
gardening.stackexchange.com/users/7393/Elaina Blodgett
I've heard of a chemical called "Milky White" that is supposed to work awesome without killing the grass.
Old housemates used it on their yard with success.
gardening.stackexchange.com/users/14018/Wendy
It's all over our community gardens.
It goes to seed really easily :(
It's easy to pull out but as you said, it keeps coming back.
I like some of your solutions!
gardening.stackexchange.com/users/21134/Junebugapril
We have  ground ivy  (creeping charlie) that is trying to take over our back yard.  We are against using pesticides, as we have a young son.  Are there natural/organic ways of beating it (other than just painstakingly pulling it all out)?
What are some natural ways of killing ground ivy (creeping charlie)?
gardening.stackexchange.com
<weed-control><organic>
gardening.stackexchange.com/questions/224
gardening.stackexchange.com/users/138/David Oneill
You could hire someone to pull it up by hand. You could also spray it with vinegar or increase the pH by adding lime. Another possibility is to let it grow tall and then mow it. You could kill it by suffocating it from light and heating it naturally or you could just hire  someone else like a neighborhood kid to pull it out. You need to keep your grass healthy to help combat ground ivy. There have been studies done using Borax to kill ground ivy with some positive results. 20% horticultural vinegar using a pump sprayer has also shown to be effective and raising the pH with lime, spraying with a bacterial based compost tea and adding calcium and magnesium will help.  Or you can simply let it grow a foot tall and mow it off at one inch.
You could kill it by suffocating it from light and heating it naturally or you could just hire  someone else like a neighborhood kid to pull it out. You need to keep your grass healthy to help combat ground ivy. There have been studies done using Borax to kill ground ivy with some positive results. 20% horticultural vinegar using a pump sprayer has also shown to be effective and raising the pH with lime, spraying with a bacterial based compost tea and adding calcium and magnesium will help.  You can let it grow a foot tall and mow it off at one inch. You can let it grow a foot tall and mow it off at one inch.