It is possible to way I'm able to, instantly, publish my Matlab output to some website.

Let us say for instance, I needed to show time during the day. I'd set Matlab to operate a forloop to include 1 second, every 1 second. Could I continuously display that output online in some way?

Yes.

Conntacting an internet site is just uploading the html file with a specific directory.

So, simply employ Matlab's ftp API to upload your file, passing on the dir path around the server in which you want the file written/submitted to.

See http://world wide web.mathworks.com/help/techdoc/ref/ftpclass.html

Create an html file from matlab, write it to disk, after which send it towards the server while using above API