How would you normally add an HTML file to your web server? Would it be (S)FTP, a file share, a git repo, Webdav, upload via a CMS?
The options your server offers can affect what approach can be taken.
If a file share, this should be possible natively I think.
If SFTP, you could automate via Secure Shellfish for example.
If Git, you,might automate via Working Copy, or using my own Drafts helper functions for Github.
If WebDAV is a native action step.
A CMS could work if it also has an API. E.g. WordPress, though there happens to be native support for this anyway.