They can't impeach him for not being born in the USA, but they can impeach him for changing laws without congressional approval and picking and choosing what parts of the law he wants to enforce.

The ACA might be called "Obamacare", but that doesn't give him the right to delay the corporate mandate for a year, exempt the big unions and tell insurance companies to extend policies that don't meet the requirements of the law. That goes beyond "executive privilege" and he has overstepped his authority here.

Congress has already started the investigations.