What you would need to do is apparently called "focus stealing" from my web searches.
At least as far as Windows is concerned, there does not seem to be a reliable way to do this from the browser alone.
I just googled "focus stealing" (which is what the JavaScript only solution would need to do to get this done) and found many answers showing that, though theoretically possible, depending on the configuration of Windows stealing the focus away from Skype by the browser would probably not work in the majority of cases.
The complaints in the Google links are numerous and some answers conflict, but it looks like reliably "stealing the focus" back to the browser is not going to be supported.
This is a good thing though, if you think about it - I do not personally want just any old JavaScript program running in my browser to change my focus from what I am working on back to the browser willy nilly - this would be a very annoying behavior for a web page to be able to do at best, making my system useless at worst.
If you could do it in this case using some methodology allowed in a browser, so could anyone else - even malevolent websites.
The best answer is to never let the focus leave the browser, but I have no idea how to do that in your specific case. Perhaps whatever means you are using to launch Skype may have an option or something to launch it in the background or whatever, never changing the focus.
I did not hit on specific links pertaining to Apple OSes, Linux or mobile OSes, but I have a feeling the same concerns and limitations apply for those as well.
Here are some of the links on the Google search (and sorry about the bad news for your needs):
Microsoft Answers Forum Post
Focus stealing is evil
http://pcsupport.about.com/od/windowsxp/ht/stealingfocus02.htm