Instalar APK a partir da sua aplicação - Android

File apk = new File( Environment.getExternalStorageDirectory() + "/Copas.apk"); 
Intent it = new Intent(Intent.ACTION_VIEW);  
it.setDataAndType(Uri.fromFile(apk),  
  "application/vnd.android.package-archive");  
startActivity(it);  

Fonte: http://nglauber.blogspot.com/2012/02/android-dicas-6.html