Download APK using DownloadManager and start installation automatically - Android 7.0 Nougat compatible

DownloadManagerActivity.java

public void downloadUpdate() {
        DownloadManager dm = (DownloadManager) getSystemService(DOWNLOAD_SERVICE);
        String destination = Environment.getExternalStoragePublicDirectory(Environment.DIRECTORY_DOWNLOADS) + "/";
        String fileName = "your_app.apk";
        destination += fileName;
        final Uri uri = Uri.parse("file://" + destination);

        File file = new File(destination);
        if (file.exists())
            file.delete();

        DownloadManager.Request request = new DownloadManager.Request(
                Uri.parse(getIntent().getStringExtra("url")));
        request.setDestinationUri(uri);
        dm.enqueue(request);

        final String finalDestination = destination;
        final BroadcastReceiver onComplete = new BroadcastReceiver() {
            public void onReceive(Context ctxt, Intent intent) {
                if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.N) {
                    Uri contentUri = FileProvider.getUriForFile(ctxt, BuildConfig.APPLICATION_ID + ".provider", new File(finalDestination));
                    Intent openFileIntent = new Intent(Intent.ACTION_VIEW);
                    openFileIntent.addFlags(Intent.FLAG_GRANT_READ_URI_PERMISSION);
                    openFileIntent.addFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP);
                    openFileIntent.setData(contentUri);
                    startActivity(openFileIntent);
                    unregisterReceiver(this);
                    finish();
                } else {
                    Intent install = new Intent(Intent.ACTION_VIEW);
                    install.setFlags(Intent.FLAG_ACTIVITY_CLEAR_TOP);
                    install.setDataAndType(uri,
                            "application/vnd.android.package-archive");
                    startActivity(install);
                    unregisterReceiver(this);
                    finish();
                }
            }
        };
        registerReceiver(onComplete, new IntentFilter(DownloadManager.ACTION_DOWNLOAD_COMPLETE));
    }

Karlos said the solution of your problem. You have to add REQUEST_INSTALL_PACKAGES to your androidManifest.xml. Without this permission, .apk files are not clickable.

android Q does not expose getExternalStoragePublicDirectory() due to security concern, so it is requested to replace 'getExternalStoragePublicDirectory()' with 'getExternalFilesDir()' for new/old version of android for downloading (i.e. writing) and reading file