First off, gas prices usually move in random manner. "We do not attempt to predict," states Avery Ash, manager for AAA. He explains how the crisis in Egypt made them rise, and how the Japan earthquake made them decline. In addition, there are many other factors that force people drive more or less. Driver miles hit a peak in 2005 and 2006 of around 3 trillion. Then mileage decreased during the recession to close to 2.9 trillion and came storming back in 2010 to around normal levels. Insurers don't adapt quickly to altering riving habits but rather base their prices on more solid numbers.
"We decrease rates because of claims experience," says Dick Luedke, an employee for State Farm. Obviously, if claims go down, then so it the rates. However, claims are generally unpredictable. If there are less cars on the road then there may be fewer accidents, but those accidents could be more fatal.
What to take from the blog? The answer to the question in the title is no. Car insurance prices cannot be predicted given only one factor such as rising gas prices. There are many other factors to take into account.