Full screen option not available when loading YouTube video in WebView

iFrame is an option but you can try this

Android's WebView and WebChromeClient class extensions that enable fully working HTML5 video support

VideoEnabledWebView

I have not try this yet but hope will help to you.

Do following changes in your java file:

view.setWebViewClient(new Browser());
view.setWebChromeClient(new MyWebClient());

---

and add this 2 class that is class Browser and class MyWebClient in java file

class Browser
        extends WebViewClient
{
    Browser() {}

    public boolean shouldOverrideUrlLoading(WebView paramWebView, String paramString)
    {
        paramWebView.loadUrl(paramString);
        return true;
    }
}

public class MyWebClient
        extends WebChromeClient
{
    private View mCustomView;
    private WebChromeClient.CustomViewCallback mCustomViewCallback;
    protected FrameLayout mFullscreenContainer;
    private int mOriginalOrientation;
    private int mOriginalSystemUiVisibility;

    public MyWebClient() {}

    public Bitmap getDefaultVideoPoster()
    {
        if (MainActivity.this == null) {
            return null;
        }
        return BitmapFactory.decodeResource(MainActivity.this.getApplicationContext().getResources(), 2130837573);
    }

    public void onHideCustomView()
    {
        ((FrameLayout)MainActivity.this.getWindow().getDecorView()).removeView(this.mCustomView);
        this.mCustomView = null;
        MainActivity.this.getWindow().getDecorView().setSystemUiVisibility(this.mOriginalSystemUiVisibility);
        MainActivity.this.setRequestedOrientation(this.mOriginalOrientation);
        this.mCustomViewCallback.onCustomViewHidden();
        this.mCustomViewCallback = null;
    }

    public void onShowCustomView(View paramView, WebChromeClient.CustomViewCallback paramCustomViewCallback)
    {
        if (this.mCustomView != null)
        {
            onHideCustomView();
            return;
        }
        this.mCustomView = paramView;
        this.mOriginalSystemUiVisibility = MainActivity.this.getWindow().getDecorView().getSystemUiVisibility();
        this.mOriginalOrientation = MainActivity.this.getRequestedOrientation();
        this.mCustomViewCallback = paramCustomViewCallback;
        ((FrameLayout)MainActivity.this.getWindow().getDecorView()).addView(this.mCustomView, new FrameLayout.LayoutParams(-1, -1));
        MainActivity.this.getWindow().getDecorView().setSystemUiVisibility(3846);
    }
}