In an exclusive interview with ABC News to be broadcast on Friday, the 82-year-old king told veteran reporter Barbara Walters that the issue in the conservative Muslim country requires patience.
"I believe strongly in the rights of women - my mother is a
woman, my sister is a woman, my daughter is a woman, my wife is a woman," he said.
"I believe the day will come when women drive," the king said in his first television interview since acceding to the Saudi throne after the death of his half-brother King Fahd on 1 August.
King Abdullah noted that women drive in Saudi Arabia's deserts and in rural areas.
"The issue will require patience," he said. "In time, I believe
it will be possible. And I believe patience is a virtue."
He said one reason he gave the exclusive interview was because Walters is a woman.
"Our people are just now beginning to open up to the world and I believe that with the passing of days in the future everything is possible," he said.
The Saudi king also said he was working to trim high oil prices. He also vowed to eliminate al-Qaida and its "work of the devil".
Abdullah said Middle East countries, including Iran and his own, should not acquire nuclear weapons and urged Iran not to become an "obstacle" to peace in Iraq.
"Without a doubt, we have benefited financially [from high oil prices], but we believe that the damage to other countries is tremendous and we don't believe prices should be at this level," the king said, according to excerpts of the interview, to air on Friday.
Asked what the kingdom could do to calm prices, Abdullah said: "We are trying and we continue to try. We have increased oil production to over 10 million barrels a day."
He expressed little concern Saudi wells might run dry, citing predictions oil supplies will last more than 70 years.
In the interview, Abdullah spoke strongly against extremism and the 11 September 2001 attacks by al-Qaida, which fractured US-Saudi relations because most of the hijackers came from the kingdom.
He called al-Qaida "madness and evil. It is the work of the devil", he said, and promised to fight "30 years if we have to until we eliminate this scourge".
Abdullah said differences over Iraq, Afghanistan and the Israeli-Palestinian conflict have soured Saudi views of America, but "we are always loyal to our friends, including the United States".