Yes but domains must be purchased from a central registrar, who is vulnerable to government regulation.
But you wouldn't really even need to move any website. The address of a website is originally numerical (the IP address), the domain name is only a service that makes addresses easier to remember for us humans. For example
http://abandonia.com/
points to
http://94.247.169.19/
But you can access a website by its IP address directly--try clicking on the second link, and the result will be the same as the link above.
When we (usually) enter the domain name instead of the IP address, we contact a third-party server that translates the former into the later, so the web server can be located. These third party (DNS) servers are one of the things this law wants to regulate.
But as I said, they're not even necessary. If they blocked a website, people would still be able to access it by its numeric IP address instead of its human-friendly domain name. That may be too much to handle for grampa or granny if they blacklisted the website where they check the weather, but computer pirates won't mind it at all.
The other tool they plan to use is to force any website located in the US, specially search services (Google etc.), not to show any link to infringing websites. This would be devastating for many commercial websites and any ones that are not widely known. In the case of widely known sites (for example thepiratebay.org), again the user community would know the link, IP if necessary.
But be sure that if they get away with this, country-wide firewalling is absolutely sure to come next. Clearly, once it's morally justified to bar access to some given content, once it's being done, but these mechanisms of doing so are so flawed, the next logical step is to make it work right: the solution is the Great Firewall of China.
In the case of the USA it might prove impossible however. Even this law now is so clearly unconstitutional. Take Wikipedia, it has an article about thepiratebay.org. Can they be forced to amend it to remove any links in it? I don't think anything could go more clearly against the First Amendment (freedom of expression). This is just a clear example, any website could claim the same. Actually this should make it impossible for this law to come into existence, even if the Congress passes it. A recent, far more difficult
case showed that the US Supreme Court is still serious about the Constitution.
Actually a Great Firewall wouldn't be (at least so clearly) unconstitutional.