I also did a clean install on a different website on the same server with the same results. I can not access the component from the backend I only get a white page with the error:
Cache refresh not detected

All the Joomla template is missing.

I am also attaching part of my Error log file. I removed duplicate error messages, and only included the error messages that contained 'zweather'.

Frustrating! I also submitted a message to Zweather about this problem. I just hope I get a reply...

Looking at your log file, it appears your site may not be able to connect to the weather.gov web site. The site needs to be able to download the XML files from this site. See if your hosting provider has blocked outbound connections from your server. One way to tell is to see if you get the news feed from Z Content in ZWeather's control panel.

Unable to open the zweather control panel.

I have attached the html source of the page that I get when I click on
Components
ZWeather
from the back end. Looking at the page the last line of code is:
<font color='red'>Cache refresh not detected</font><br/>

To me it looks like the page is not complete.

Also I went to the server and ran:
wget weather.gov
The page was saved without any issues, so the server does look to have access to the weather.gov site.

Also check to see if the PHP option allow_url_fopen is enabled. You can check for this under the "Help -> System Info -> PHP Information" menu.

Yes 'allow_url_fopen' is 'on'.