Technically you can delete the Windows partition and it would be fine. However you should backup your files in it if your Windows partition was all about C:\ drive. If you are gonna backup anyway, just do the backup and format the whole disk and start with a clean slate. It’s better if your disk is not formatted as NTFS.
If you have D:\ drive and want to delete C:\ drive and only keep D:, you should also delete System Reserved and EFI partitions as well. You can create another partition with this place but that would make 2 partitions in the disk. Normally Gparted (or any other partition program) can merge partitions without you losing data but I have no idea if they can do this with NTFS as well. So, I would go with the first option here. Backup everything and format the disk with a Linux file system. If you are gonna use it for mostly media files, XFS could be a fine choice. If you want it as a general separate disk, go with ext4.
If you want to learn more, I can answer your questions or link you Wiki pages.
I feel like this is somehow related to Windows 10 not being really shut down when you shutdown. Try restarting Windows, and while before it gets pass BIOS, interrupt and shut down there. Then replace the drives and try to boot Linux again.