Yes earwigs do sometimes eat ants.
I've had earwigs in my yard and I have witnessed the earwigs eating some of the ants that were near them so yes earwigs do eat ants.
Earwigs also eat green leafs such as lettuce, leaves from trees etc and earwigs can be pretty damaging to gardens and trees.
Also do not pickup an earwig with your bare hands as earwig pincers do hurt if they pinch you with them.
I have been pinched by an earwig pincer and it does hurt and is very painful but the pain subsides within about 30 minutes or so.
I would highly recommend to not ever pick up an earwig unless you're wearing thick gloves because they can and do pinch you with their earwig pincers.
The earwig pincers can break skin too and cause bleeding.
That happened to me and it's because I picked up an earwig with my bare hands and didn't wear gloves.
Now I wear gloves when picking up or touching an earwig.
Earwigs do not bite but they do pinch you.