If women earn less for doing the same work that a man does (.77 to the dollar according to Obama), why do employers not do their best to hire only women?

First off, the $.77 cents to the dollar stat is misleading. If you don't control for things like career choice, time in the workforce, experience, etc., then it's meaningless. Men tend to dominate in technical fields as well as ones with hazard pay, like mining and welding. Also, many women leave the workforce to have children for a period of time and then return with less experience than their male peers. Lastly, the wage gap is not monolithic. In some regions, young women get paid more than young men by large margins. According to this article, in Memphis and Atlanta women make 20% more than their male counterparts.

As someone in IT, I can tell you that female programmers with less experience will get hired over male programmers. Also, they commonly are offered more money. This is primarily to do with scarcity and the desire to have a more balanced workplace. That being said, there is a slight wage gap overall between genders for workers of equivalent experience and education.

A major component of the wage gap is that many older women in the workforce started out before social norms had changed. New studies show that really it's starting salaries that determine trajectories. In the next 20 years, we are likely to see a sort of see-saw effect of women outpacing men in many areas due to attaining much higher levels of education than their male counterparts. However, this will also then create the opposite imbalance and make qualified men scarce. We actually see this in nursing now. Male nurses are harder to find and thus are offered more money. Female programmers are harder to find, so they are offered more money.

That being all being said, there are many different factors that contribute to an actual wage gap and a perceived wage gap and it really depends on things like region, industry, etc.