No
oil has been exported from the North Slope since 2004. The total US exports of crude has been a maximum of 77,000 barrels per day in 2012 and all of it to Canada. Now we import almost 3 million barrels per day from Canada so I don't think sending them 77,000 barrels per day is much of a problem.
In some cases it would be better to export crude where the transportation cost is cheaper than importing a like amount of crude. There have been cases this was the case but the politics prevented it from happening.
By the way the major problem with California gasoline price is that it's a specific requirements that most refineries are not set up to produce so when you have a problem at any of those that produce California gasoline you can't import gas from other sources. The changeover from Summer to Winter gas and back just makes the problem worse