Very little is known about Kearney. He was born in Kent County, Delaware, but his birthdate isn't known. He was admitted to the bar of New Castle County in 1784, and practiced law in Dover until his death there in 1791.
He was the son of Edward Kearney and Rebecca Dyre (or Dyer)(d. abt. 1795). Rebecca was a great granddaughter of the Quaker martyr Mary Dyer.