WebView uses flash.media.StageWebView internally. I'm not sure if StageWebView supports PDFs. The StageWebView docs say that StageWebView uses AIR's WebKit engine by default on desktop, so maybe.
According to the flash.html.HTMLLoader docs, HTMLLoader (which is a more powerful way to display a browser in your app than StageWebView) can display PDFs, but only if the Adobe Reader plugin is installed on the system. That may indicate that StageWebView can display a PDF on desktop if you have the plugin. However, that's not necessarily guaranteed since StageWebView is designed to be more restricted because it was introduced specifically for mobile.