Unfortunately, a deleted Xbox 360 profile cannot be recovered.
Why It's Permanent
Local Storage Only: Profiles are saved to the console's hard drive or memory unit. No cloud sync existed for Xbox 360 profiles.
No Server Backup: Xbox Live stores your Gamertag, achievements, and licenses, but not the profile file itself (save data, settings, local content).
Deletion = Wipe: Choosing "Delete Profile Only" or "Delete Profile and Items" removes the file permanently.
What You Can Do
Create a New Profile: Sign in with the same Microsoft account; achievements and licenses remain, but game saves are gone.
Contact Xbox Support: They cannot restore the profile, but can verify if any cloud saves exist for specific titles (rare for 360).
Prevent Future Loss: Use Settings > System > Storage > Cloud Saved Games (requires Gold) on Xbox One/Series to back up saves automatically. For 360, manually copy saves to a USB drive regularly.