King led a successful bid to oust the previous board at Ibrox in March, but had to wait for clearance from the SFA given his convictions over tax arrangements in South Africa.
He was also a member of the Rangers board before the club's administration and liquidation in 2012.
The SFA said in a statement that it was satisfied King had passed the necessary tests "after receiving specialist legal advice in Scotland and South Africa".
King, who reportedly has a personal fortune in the region of 250 million pounds ($387.7 million), told the Rangers website he was "delighted" at the announcement and would now "turn (his) attention fully to the task of re-building Rangers."
Supporters will see the ruling as a major step forward after more than three years in the doldrums.
The record 54-times Scottish champions had to re-enter the game in the lowest professional league following their liquidation in 2012 and they have climbed back to the second tier.
They face Hibernian on Wednesday in the first leg of the Scottish Premiership playoff. The winners go forward to face Motherwell for a place in the top flight.
(Reporting by Douglas Beattie; Editing by Ed Osmond)
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