No, this is not possible for security reasons. You can initiate the installation but it is up to the user to decide.
However if the phone is rooted there is a way to do this programmatically but this is not an option for most of the apps out there. Code is similar to this:
public ApplicationManager(Context context) throws SecurityException, NoSuchMethodException {
observer = new PackageInstallObserver();
pm = context.getPackageManager();
Class<?>[] types = new Class[] {Uri.class, IPackageInstallObserver.class, int.class, String.class};
Class<?>[] uninstalltypes = new Class[] {String.class, IPackageInstallObserver.class, int.class};
method = pm.getClass().getMethod("installPackage", types);
uninstallmethod = pm.getClass().getMethod("deletePackage", uninstalltypes);
}
public void uninstallPackage(String packagename) throws IllegalArgumentException, IllegalAccessException, InvocationTargetException {
uninstallmethod.invoke(pm, new Object[] {packagename, observer, 0});
}
public void installPackage(Uri apkFile) throws IllegalArgumentException, IllegalAccessException, InvocationTargetException {
method.invoke(pm, new Object[] {apkFile, observer, INSTALL_REPLACE_EXISTING, null});
}
More details here:
install / uninstall APKs programmatically (PackageManager vs Intents)
https://paulononaka.wordpress.com/2011/07/02/how-to-install-a-application-in-background-on-android/