You can download the new APK file to SD card, then call this to install it:
Intent shareIntent = new Intent(Intent.ACTION_VIEW);
shareIntent.setDataAndType(Uri.fromFile(new File("path-to-APK-file")),
"application/vnd.android.package-archive");
try {
context.startActivity(shareIntent);
} catch (Throwable t) {
// handle the exception here
}
There is only one thing not automatic: the final step. The system will ask the user to confirm installation.
About the MIME type of APK files, here's the wiki page.