Use wp_remote_get() with a private or password protected page?

Is there any way to use the wp_remote_get() function to access a WordPress page that is either set to Private -OR- password protected (using the native password option in the post visibility settings)? The page I need to access is on the same site.

I am using this function in a cron event to access this page, but I don't want this page to be accessible by the public or search bots. Googlebot is ignoring my "noindex" on the page.

Thank you!