I am assuming the domain registrar and hosting company are both US based.
Personal opinion, not legal advice:
First and foremost: I assume they meant trademark, and not copyright? If they do not have a trademark, unless you have strikingly similar content on your site, you don't have to worry about anything.
If they really used the word "copyright":
1. Make sure your site is clean. All content and images should be unique.
2. If you are using any copyrighted terms, provide attributions on your web site.
3. Respond to them, stating (1) and (2) and forget about it.
If this is about trademarks:
Since you have a developed site (based on your post), they cannot accuse you of bad faith registration, or say that you do not have any "legitimate interests". The only point they can argue about is the trademark/service mark.
If "ABC" is a generic term, then you are probably safe.
If "ABC" is a trademarked term in their country, but not here, and if your site's use of "ABC" is in a different context, then also, you should be alright.
If "ABC" is trademarked in US by **any other** company, then point
it out to them. Their case will be weak in this case.
If they have a trademark/patent for "ABC" within USA, and if your site very clearly targets markets outside USA, you can point that fact out to them in your response.
I recently bought a site with the domain abcfree.com (not real). The company ABC is claiming that this infringes upon their copyright and I set a deadline for me to comply with turning the domain over.
Does anyone have any experience with this? The company is big (there's a good chance you've heard of them) and not based in the US so my greatest fear is that they would pursue a lawsuit in their country.
Honestly I would just raise the white flag and turn the domain over but the site really needs the domain.