Formatting a hard drive usually doesn’t erase the data, it just marks the space as available for writing in the future. If you haven’t written (much) to that drive since formatting it, there’s a good chance your private key is still in there. There’s freely available recovery software if you know what you’re doing, or you can take the drive to a reputable data recovery company (reputable because you don’t want them to make a copy of your private key and steal your bitcoins).
No, you can’t recover those bitcoins without a private key.
