"doubts whether or not" is correct and usable in written English.
You can use it when expressing uncertainty or indecision. For example, "She had her doubts whether or not she should follow their suggestion."
"You doubt whether or not they're trying to create a nuclear bomb?" Mr. Graham persisted.
Why is Mr. Woodward "doubting" whether or not Mr. Felt is "competent" to tell his story?
Weeks such as we've had in Iraq make some doubt whether or not we're making progress.
"People doubted whether or not I had the stamina," Mr. Thompson said to a packed room of supporters at a hotel here, adding that he has been "shaking hands like I've never shaken before".
"There is no dispute there were other female family members present at the time, the only real issue is to determine beyond reasonable doubt whether or not each of the girls had their clitoris' mutilated by, there is a concern whether or not there was any actual bodily harm inflicted on the girls," he said.
You'll recall that fancy-pants analysts doubted whether or not Black Ops would reach the same kind of penetration that its predecessor, Modern Warfare 2, did.
Until he does those things, people will doubt whether or not he really won Ohio.
There were some who doubted whether or not Gardena Serra could actually play with the likes of the many formidable opponents in the re-aligned, seven-team Mission League.
This is why one survivor once told me that after she reported her sexual assault to the police, she thought to herself, "I wish I was murdered instead so nobody had to doubt whether or not I was really raped".
Americans today are pretty sure that the government's postal service will deliver their mail on time, but they have every reason to doubt whether or not Chrysler, GM, AIG and Bear Stearns will be doing business tomorrow.
