I don't believe (but haven't recently checked) there is any mandatory insurance requirement in the USA for private aircraft operation. HOWEVER, the USA is the litigation capital of the world, so operating an uninsured expensive aircraft would be madness. Also anyone who has an interest in the aircraft (i.e. the trust company, lender, etc.) will certainly demand insurance.
The insurance company will almost certainly demand a high minimum PIC time, a type specific course (i.e. FlightSafety type course) and a certain minimum number of hours supervised by a pilot (could be instructor or 'just' an ATPL with significant time on type). This is one of the reasons a type rating is less important in the US - you wind up with nearly the same training due to the insurance requirements!