Скачивание файлов на телефоне из webview

mWebView.setDownloadListener(new DownloadListener() {
    public void onDownloadStart(String url, String userAgent,
                String contentDisposition, String mimetype,
                long contentLength) {
        Intent i = new Intent(Intent.ACTION_VIEW);
        i.setData(Uri.parse(url));
        startActivity(i);
    }
});

Попробуйте это. Пройдя много постов и форумов, я нашел это.
 
mWebView.setDownloadListener(new DownloadListener() {       
 
    @Override
    public void onDownloadStart(String url, String userAgent,
                                    String contentDisposition, String mimetype,
                                    long contentLength) {
            DownloadManager.Request request = new DownloadManager.Request(
                    Uri.parse(url));
 
            request.allowScanningByMediaScanner();
            request.setNotificationVisibility(DownloadManager.Request.VISIBILITY_VISIBLE_NOTIFY_COMPLETED); //Notify client once download is completed!
            request.setDestinationInExternalPublicDir(Environment.DIRECTORY_DOWNLOADS, "Name of your downloadble file goes here, example: Mathematics II ");
            DownloadManager dm = (DownloadManager) getSystemService(DOWNLOAD_SERVICE);
            dm.enqueue(request);
            Toast.makeText(getApplicationContext(), "Downloading File", //To notify the Client that the file is being downloaded
                    Toast.LENGTH_LONG).show();
 
        }
    });

2019-02-18 15:11:30.104 3565-5486/android.process.media E/DatabaseUtils: Writing exception to parcel
    java.lang.SecurityException: No permission to write to /storage/emulated/0/Download/Разъяснения.docx: Neither user 10085 nor current process has android.permission.WRITE_EXTERNAL_STORAGE.
        at android.app.ContextImpl.enforce(ContextImpl.java:1812)
        at android.app.ContextImpl.enforceCallingOrSelfPermission(ContextImpl.java:1840)
        at android.content.ContextWrapper.enforceCallingOrSelfPermission(ContextWrapper.java:776)
        at com.android.providers.downloads.DownloadProvider.checkFileUriDestination(DownloadProvider.java:784)
        at com.android.providers.downloads.DownloadProvider.insert(DownloadProvider.java:592)
        at android.content.ContentProvider$Transport.insert(ContentProvider.java:265)
        at android.content.ContentProviderNative.onTransact(ContentProviderNative.java:152)
        at android.os.Binder.execTransact(Binder.java:731)
2019-02-18 15:11:30.105 8284-8284/v1.v1 D/AndroidRuntime: Shutting down VM
    
    
    --------- beginning of crash
2019-02-18 15:11:30.106 8284-8284/v1.v1 E/AndroidRuntime: FATAL EXCEPTION: main
    Process: v1.v1, PID: 8284
    java.lang.SecurityException: No permission to write to /storage/emulated/0/Download/Разъяснения.docx: Neither user 10085 nor current process has android.permission.WRITE_EXTERNAL_STORAGE.
        at android.os.Parcel.createException(Parcel.java:1942)
        at android.os.Parcel.readException(Parcel.java:1910)
        at android.database.DatabaseUtils.readExceptionFromParcel(DatabaseUtils.java:183)
        at android.database.DatabaseUtils.readExceptionFromParcel(DatabaseUtils.java:135)
        at android.content.ContentProviderProxy.insert(ContentProviderNative.java:476)
        at android.content.ContentResolver.insert(ContentResolver.java:1587)
        at android.app.DownloadManager.enqueue(DownloadManager.java:1022)
        at v1.v1.MainActivity$2.onDownloadStart(MainActivity.java:105)
        at com.android.webview.chromium.Ap.onDownloadStart(WebViewContentsClientAdapter.java:492)
        at org.chromium.android_webview.AwContentsClientCallbackHelper$MyHandler.handleMessage(AwContentsClientCallbackHelper.java:25)
        at android.os.Handler.dispatchMessage(Handler.java:106)
        at android.os.Looper.loop(Looper.java:193)
        at android.app.ActivityThread.main(ActivityThread.java:6669)
        at java.lang.reflect.Method.invoke(Native Method)
        at com.android.internal.os.RuntimeInit$MethodAndArgsCaller.run(RuntimeInit.java:493)
        at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:858)

<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"/>