Well, I don’t think you can do this with cron.
Browser sends request to server and gets response. It isn’t connected permanently to your server. So if you want to refresh page which is already displayed in browser, you have to force browser to send request for this page one more time (do refresh).
Another way would be placing some JS on your page. It could send AJAX request every few minutes (or so) to your server and check if there is anything new to show and process them in any way you’d like.