A website can be protected as a wordmark. However, the point is that its name must reflect the product it sells or the service it provides. The individual website name must be implemented into something tangible that the customer can buy or use. It is not possible to register just a domain name without that name carrying your brand of products or services.
This question is, therefore, a bit tricky, and the answer is not easy. If you own a domain name, it does not guarantee you the right to claim legal protection for it as a trademark. The same is true in reverse; if you have registered a trademark, it does not automatically guarantee you can have it as a domain if someone is using this name for their website.
However, if you have a registered wordmark and the domain owner misuses your name on a website and promotes under your name, it is possible to legally challenge them and enforce your rights.