Short answer: it might not be possible. Formatting a disk (or rather creating a filesystem) will write around the disk and break the previous filesystem. Installing a new operating system will write a lot of data on top of the previous one.
All of this makes it very hard to get any data back. You can try with data recovery tools, they might find fragments of filesystem data and might not care that there is a different filesystem on the disk right now. But don't get your hopes up because depending on how much data was written it might have corrupted blocks from here and there in many files.
Unless you have backups (which I assume you don't, since asking about this) it will be slow, painful and maybe expensive operation involving professional data recovery companies.
But do not operate the Linux installation on the disk. Every time you use it more data will be destroyed.