can a virus execute by itself?

"Yes, in the context of the browser, since unintentionally you're executing the page without clicking on anything. Such viruses are capable of downloading themselves to your hard disk without your cooperation.

The propagation vector here can be JavaScript, Java, ActiveX, Flash and other plugins. Many such attacks are carried out through cross-site scripting.

You can find lots of information about Web attacks on the site of the popular Firefox extension NoScript."