Were you able to resolve this? I wanted to check the default browser on the user's device before launching a URL.
Per my understanding, this is not possible or at least shouldn't be possible because it would violate Apple's user privacy since you as a developer would know that certain application is installed on the device.
Think about it in such a way: if Google is not doing it in their applications to detect whether Chrome is set as default browser why would I do that?
You may look into Firefox for iOS source code for more details if you need to display any message to user asking him/her to set your browser as default to get some inspiration but I am pretty sure that they and other browser-implementing guys haven't found any meaningful answer to your question.
FireFox iOS sources: https://github.com/mozilla-mobile/firefox-ios
Chromium sources: https://chromium.googlesource.com/chromium/src.git/+/master/ios/
