Thanks for such a quick response Chris! I've managed to resolve it (sort of) by making the following changes -
1. Set the base url under website settings to https://<sitename>/ instead of just the http://<sitename>
2. For the Innovation theme, edited the header.inc.php and changed url http://fonts.googleapis.com/css?family=Yanone+Kaffeesatz' to https://fonts.googleapis.com/css?family=Yanone+Kaffeesatz'
3. Edited the sidebar.inc.php file and changed the URLs - http://www.addthis.com/bookmark.php?v=250 and
http://s7.addthis.com/js/250/addthis_widget.js to https://www.addthis.com/bookmark.php?v=250 https://s7.addthis.com/js/250/addthis_widget.js respectively.
Thankfully, all 3 URLs did support the https version.
After these changes all 3 browsers displayed the https version of the with no errors.
It looks by design GS Core uses the "Website Base URL" to call resources - so now , even when I access the http version and look at the source, a lot of resources are being called from the https version. I'm wondering if this should be changed to use a relative URL.
Thanks!
Venky
Edit: Oh..if you want to get ssl certs for your website, check out startssl.com - they can do class 1 domain validation and provide ssl certs for free.
Edit 2: I'm not so sure if its the core or the theme to blame - Here are the resources that seem to be using absolute URLs instead of relative uri
/theme/Innovation/assets/css/reset.css
/theme/Innovation/style.css?v=3.1B
/theme/Innovation/assets/js/dd_belatedpng.js ( IE Only)
/theme/Innovation/assets/images/break.png
/theme/Innovation/assets/images/share.png
I'll try out few other themes and check.
Edit 4: All themes are calling the function get_site_url() which looks up variable $SITEURL , which is a static entry in websites.xml file. So as far as I can understand, there is no easy way to change the behavior of themes to use dynamic entries based on the accessed URL.