Yes. You don't have to be a natural-born US citizen, though; naturalized citizens (like, for example, Arnold Schwarzenegger) can become governors.
in the US anywhere
The current governor of South Carolina, Nikki Haley, was born in Bamberg, South Carolina and assumed office on January 12th, 2011. Did you know that Haley is currently the youngest serving governor in The United States?
There is no such thing as a governor of the US. Each state in the U.S has a governor, thus there are 50 governor's in the US. The President is the Chief Executive and his name is George W. Bush.
Richard Governor was born in Australia.
A Governor is head of a state government in the US.
Samuel Mathews - Governor - was born in 1630.
Alan Burns - governor - was born in 1887.
William Markham - Governor - was born in 1635.
John Cranston - governor - was born in 1625.
Hiroshi Saitō - governor - was born in 1957.
Richard Bulkeley - governor - was born in 1717.
Lewis Morris - governor - was born in 1671.