If you mowed the lawn, and you have a grass collector on your mower, then I think that will have taken care of it. The overspray won't have been strong enough, I think it will have just affected the top bits of grass which you have mowed now?
Mostly dogs eat grass which is longer. They seem to be attracted to the long strands more. I have never known a dog try to eat a mowed lawn. I don't think short grass 'does the trick' for them, when they are in a grass-eating mood.
And if any remnants of paint are now left, they will have dried so aren't likely to get on his paws.
But if you are still worried you can sacrifice a patch of lovely green grass and spread a dust-sheet over the area to be absolutely certain, and keep your mind at rest. When you take it up, that area will have turned yellow however!
I think he would probably not eat enough grass to make him sick. But dogs usually throw up after eating a lot of grass anyway, so if he did eat some of the painted grass it might just come back out pretty quick. I doubt he would like the smell or taste of the painted grass either.