With the FileSystemWatcher, no. That object only stays active for as long as the process / AppDomain is running, at the most. You could look into a permanent WMI event consumer, but honestly, all you're doing here is (partially) reinventing the wheel of file system auditing. Why not just enable auditing, and review the Security event log when you want to know what files were changed, and when?
If you need a permanent FileSystemWatcher, you're getting into .NET programming, and making a background service. The whole point of a service is to run persistent code. And as Dave points out, the event log does this already, with less overhead.