I explored web in order to solve this issue but it seems it is not possible to crawl an external website using SharePoint search that has STS based authentication and don't have Windows authentication enabled. Here are few references:
"If you are using claims-based authentication, make sure that that Windows authentication is enabled on any Web applications to be crawled." Source
"SharePoint can only crawl using a Windows NTLM authentication account. Your content source must authorize the Windows account sent as part of the crawl request in order to access the document content. Though claims authentication is supported in SharePoint 2010, the gatherer is still not a claims-aware application and will not access a content source that has claims authentication only." Source
"No, it needs NTLM/Kerberos in order to crawl the Web App." Source