During my search for a way to view JSON files in a WebBrowser on my desktop app, I encountered the same issue in Internet Explorer where it prompted me to download the file. After much perseverance, I finally found the solution.
Here's what you need to do:
Open Notepad and copy the following code:
[HKEY_CLASSES_ROOT\MIME\Database\Content Type\application/json]
"CLSID"="{25336920-03F9-11cf-8FD0-00AA00686F13}"
"Encoding"=hex:08,00,00,00
Save the document as Json.reg, then right click on the file and select "Run as administrator."
After completing these steps, you will be able to view JSON files in both IE and your Desktop WebBrowser. Enjoy!