I verify launched current activity if it's in browser or in app by comparing with current activity.
activity = driverAppium.current_activity
And then I verify if activity matches with browser activity name e.g. org.chromium.browser...
But can I verify the http response on the webpage e.g. 200 or 404?
With above test always passes even though webpage didn't load or get null response.
Can I verify with current activity and response both?
There are two ways to do it what I can think of, 

UI prospect :

Capture the screenshot of the webview with 200 response. Let's call it expectedScreen.png
Capture the screenshot of the under test response(be it 200, 400 etc.). Lets call this finalScreen.png
Compare both the images to verify/assert.

API prospect : Since the Activity suppose to be displayed shall never/rarely be changed depending on transitions between different activities on your application as designed, so verifying current activity is a less important check during the test. You can rather verify these using API calls and then(if you get proper response) look for presence of elements on screen accordingly.
Verify appium current activity response