What obligations does an employer have if a majority of workers in their workplace want union representation?
In a politically free, democratic system, workers are free to organize. That freedom, in fact, is a bedrock of American democracy, and freedom to unionize remains a measure against which Americans judge human rights in foreign countries. If American workers vote to unionize, the company in question cannot legally stop them. Those workers have the right to unionize.
The question is what can business owners who oppose unionization of their labor forces do to legally prevent such a step? The positive approach is to address those grievances held by employees that have led to discussions of unionization. If workers are agitating for fewer work hours, higher hourly pay, better health benefits, etc., then business owners have to consider comprising on those issues. Increasing costs associated with increasing pay structures, providing better health insurance coverage, and reducing productivity due to fewer hours worked would have to be passed on to consumers or to, depending upon the company's place in the manufacturing/service chain, to other businesses (which would, inevitably, be passed along to that company's clients). Savvy labor negotiators understand that, if they push too hard for too great an increase in benefits for workers, they could inadvertently (or even intentionally) force the company into bankruptcy.
A business owner who, for whatever reason, refuses to negotiate with his or her employees leaves him- or herself open to an organized effort at unionizing. That leaves the owner with the unattractive but very viable option of seeking to sell the business, or to simply shut down operations and dissolve the corporation. Often, the threat by business owners to pursue one of these options is sufficient to force labor to back down lest all of the employees find themselves unemployed. A business owner cannot fire employees for unionizing, but a business owner can sell the business or close its doors. If the workers, as pointed out above, wish to organize, they are free to do so; if a business owner wishes to respond to such a measure by selling the company (especially if the potential buyer of the company is headquartered in a foreign country with lower labor rates and standards -- a serious threat to the workers whose jobs would be shifted overseas), he is equally free to entertain that option.