tedlyc61 on "WP and plugins won't update--SSL could not be verified"

Hi,
I have a site running 4.1.1 on a GoDaddy server with PHP 5.4. Whenever I try to update WordPress or a plugin, I get the following error:

"The SSL certificate for the host could not be verified."

GoDaddy support thought that upgrading from PHP 5.3 to 5.4 might help, but no. I can get up to PHP 5.5 if my client buys a different hosting package, but I'd like to explore other solutions first, especially since I have no indication that the PHP version is a problem yet.

The site does have a certificate installed which seems to be up to date.

GoDaddy claims the server is fine and they cannot help me much beyond that because it's a manual install of WP. Great.

I recently made a clone of the site which I'm running on a development server (Dreamhost) and both WP and plugins updated just fine there.

In my research so far, some posts suggest that this issue may be caused by outdated server components, or possibly a 'missing root CA'. Or an older Open SSL version, whatever that is.

I could take the site down (not very optimal) and manually update WP to the latest version if that would solve the problem, but I'd rather not have to do that. I've also seen methodology suggested for similar-sounding issues that involves disabling the SSL verification but that doesn't sound real smart.

Any ideas? I'm out of my depth on certificates, so some language that I could use to point GoDaddy support in the right direction would be helpful, if indeed it's a server issue. Thanks!