Two options: keep the tickets, and you both go separately, and risk running into each other.

Or: refund the tickets, and go somewhere else with no ties to him.

Oh, or, I guess, a third option would be to just change the dates of one ticket, so you can both still go, but at different times. That way there's no running into each other there and ruining a perfectly good vacation.

Solo travel advice: do it! It seems scary, but it's not. It's easier than you'd think. Granted, I've never been to Iceland, but a lot of places have signs in English as well as their language, so you just have to keep a look out. Figure out what you want to see, and leave yourself time to explore. Relax. Or if you prefer more scheduled things, or are worried about getting lost, etc, do group tours. Ask your hotel or, heck, call AAA (they still exist, right?) for recommendations so you know they're legit.

Also, look online before you go for advice on traveling to the area, as far as what you'll need, what to do, what to avoid, etc. Go in with an idea of what you're doing and it won't be nearly so scary.

Take the plunge!